:: ZF_MODEL semantic presentation
scheme :: ZF_MODEL:sch 22
s22{
F1(
Variable,
Variable)
-> set ,
F2(
Variable,
Variable)
-> set ,
F3(
set )
-> set ,
F4(
set ,
set )
-> set ,
F5(
Variable,
set )
-> set ,
F6()
-> ZF-formula } :
ex
a,
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),F1(x,y)] in A &
[(x 'in' y),F2(x,y)] in A ) ) &
[F6(),a] in A & ( for
H being
ZF-formula for
a being
set st
[H,a] in A holds
( (
H is_equality implies
a = F1(
(Var1 H),
(Var2 H)) ) & (
H is_membership implies
a = F2(
(Var1 H),
(Var2 H)) ) & (
H is
negative implies ex
b being
set st
(
a = F3(
b) &
[(the_argument_of H),b] in A ) ) & (
H is
conjunctive implies ex
b,
c being
set st
(
a = F4(
b,
c) &
[(the_left_argument_of H),b] in A &
[(the_right_argument_of H),c] in A ) ) & (
H is
universal implies ex
b being
set st
(
a = F5(
(bound_in H),
b) &
[(the_scope_of H),b] in A ) ) ) ) )
scheme :: ZF_MODEL:sch 96
s96{
F1(
Variable,
Variable)
-> set ,
F2(
Variable,
Variable)
-> set ,
F3(
set )
-> set ,
F4(
set ,
set )
-> set ,
F5(
Variable,
set )
-> set ,
F6()
-> ZF-formula,
F7()
-> set ,
F8()
-> set } :
provided
E30:
ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),F1(x,y)] in A &
[(x 'in' y),F2(x,y)] in A ) ) &
[F6(),F7()] in A & ( for
H being
ZF-formula for
a being
set st
[H,a] in A holds
( (
H is_equality implies
a = F1(
(Var1 H),
(Var2 H)) ) & (
H is_membership implies
a = F2(
(Var1 H),
(Var2 H)) ) & (
H is
negative implies ex
b being
set st
(
a = F3(
b) &
[(the_argument_of H),b] in A ) ) & (
H is
conjunctive implies ex
b,
c being
set st
(
a = F4(
b,
c) &
[(the_left_argument_of H),b] in A &
[(the_right_argument_of H),c] in A ) ) & (
H is
universal implies ex
b being
set st
(
a = F5(
(bound_in H),
b) &
[(the_scope_of H),b] in A ) ) ) ) )
and E31:
ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),F1(x,y)] in A &
[(x 'in' y),F2(x,y)] in A ) ) &
[F6(),F8()] in A & ( for
H being
ZF-formula for
a being
set st
[H,a] in A holds
( (
H is_equality implies
a = F1(
(Var1 H),
(Var2 H)) ) & (
H is_membership implies
a = F2(
(Var1 H),
(Var2 H)) ) & (
H is
negative implies ex
b being
set st
(
a = F3(
b) &
[(the_argument_of H),b] in A ) ) & (
H is
conjunctive implies ex
b,
c being
set st
(
a = F4(
b,
c) &
[(the_left_argument_of H),b] in A &
[(the_right_argument_of H),c] in A ) ) & (
H is
universal implies ex
b being
set st
(
a = F5(
(bound_in H),
b) &
[(the_scope_of H),b] in A ) ) ) ) )
scheme :: ZF_MODEL:sch 100
s100{
F1(
Variable,
Variable)
-> set ,
F2(
Variable,
Variable)
-> set ,
F3(
set )
-> set ,
F4(
set ,
set )
-> set ,
F5(
Variable,
set )
-> set ,
F6()
-> ZF-formula,
F7(
ZF-formula)
-> set } :
provided
E30:
for
H' being
ZF-formula for
a being
set holds
(
a = F7(
H') iff ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),F1(x,y)] in A &
[(x 'in' y),F2(x,y)] in A ) ) &
[H',a] in A & ( for
H being
ZF-formula for
a being
set st
[H,a] in A holds
( (
H is_equality implies
a = F1(
(Var1 H),
(Var2 H)) ) & (
H is_membership implies
a = F2(
(Var1 H),
(Var2 H)) ) & (
H is
negative implies ex
b being
set st
(
a = F3(
b) &
[(the_argument_of H),b] in A ) ) & (
H is
conjunctive implies ex
b,
c being
set st
(
a = F4(
b,
c) &
[(the_left_argument_of H),b] in A &
[(the_right_argument_of H),c] in A ) ) & (
H is
universal implies ex
b being
set st
(
a = F5(
(bound_in H),
b) &
[(the_scope_of H),b] in A ) ) ) ) ) )
scheme :: ZF_MODEL:sch 102
s102{
F1(
Variable,
Variable)
-> set ,
F2(
Variable,
Variable)
-> set ,
F3(
set )
-> set ,
F4(
set ,
set )
-> set ,
F5(
Variable,
set )
-> set ,
F6(
ZF-formula)
-> set ,
F7()
-> ZF-formula,
P1[
set ] } :
provided
E30:
for
H' being
ZF-formula for
a being
set holds
(
a = F6(
H') iff ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),F1(x,y)] in A &
[(x 'in' y),F2(x,y)] in A ) ) &
[H',a] in A & ( for
H being
ZF-formula for
a being
set st
[H,a] in A holds
( (
H is_equality implies
a = F1(
(Var1 H),
(Var2 H)) ) & (
H is_membership implies
a = F2(
(Var1 H),
(Var2 H)) ) & (
H is
negative implies ex
b being
set st
(
a = F3(
b) &
[(the_argument_of H),b] in A ) ) & (
H is
conjunctive implies ex
b,
c being
set st
(
a = F4(
b,
c) &
[(the_left_argument_of H),b] in A &
[(the_right_argument_of H),c] in A ) ) & (
H is
universal implies ex
b being
set st
(
a = F5(
(bound_in H),
b) &
[(the_scope_of H),b] in A ) ) ) ) ) )
and E31:
for
x,
y being
Variable holds
(
P1[
F1(
x,
y)] &
P1[
F2(
x,
y)] )
and E32:
for
a being
set st
P1[
a] holds
P1[
F3(
a)]
and E33:
for
a,
b being
set st
P1[
a] &
P1[
b] holds
P1[
F4(
a,
b)]
and E34:
for
a being
set for
x being
Variable st
P1[
a] holds
P1[
F5(
x,
a)]
deffunc H1( Variable, Variable) -> set = {a1,a2};
deffunc H2( Variable, Variable) -> set = {a1,a2};
deffunc H3( set ) -> set = a1;
deffunc H4( set , set ) -> set = union {a1,a2};
deffunc H5( Variable, set ) -> Element of bool (union {a2}) = (union {a2}) \ {a1};
definition
let H be
ZF-formula;
func Free c1 -> set means :
Def1:
:: ZF_MODEL:def 1
ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),{x,y}] in A &
[(x 'in' y),{x,y}] in A ) ) &
[H,it] in A & ( for
H' being
ZF-formula for
a being
set st
[H',a] in A holds
( (
H' is_equality implies
a = {(Var1 H'),(Var2 H')} ) & (
H' is_membership implies
a = {(Var1 H'),(Var2 H')} ) & (
H' is
negative implies ex
b being
set st
(
a = b &
[(the_argument_of H'),b] in A ) ) & (
H' is
conjunctive implies ex
b,
c being
set st
(
a = union {b,c} &
[(the_left_argument_of H'),b] in A &
[(the_right_argument_of H'),c] in A ) ) & (
H' is
universal implies ex
b being
set st
(
a = (union {b}) \ {(bound_in H')} &
[(the_scope_of H'),b] in A ) ) ) ) );
existence
ex b1, A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,b1] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is_equality implies a = {(Var1 H'),(Var2 H')} ) & ( H' is_membership implies a = {(Var1 H'),(Var2 H')} ) & ( H' is negative implies ex b being set st
( a = b & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = union {b,c} & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = (union {b}) \ {(bound_in H')} & [(the_scope_of H'),b] in A ) ) ) ) )
uniqueness
for b1, b2 being set st ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,b1] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is_equality implies a = {(Var1 H'),(Var2 H')} ) & ( H' is_membership implies a = {(Var1 H'),(Var2 H')} ) & ( H' is negative implies ex b being set st
( a = b & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = union {b,c} & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = (union {b}) \ {(bound_in H')} & [(the_scope_of H'),b] in A ) ) ) ) ) & ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{x,y}] in A & [(x 'in' y),{x,y}] in A ) ) & [H,b2] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is_equality implies a = {(Var1 H'),(Var2 H')} ) & ( H' is_membership implies a = {(Var1 H'),(Var2 H')} ) & ( H' is negative implies ex b being set st
( a = b & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = union {b,c} & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = (union {b}) \ {(bound_in H')} & [(the_scope_of H'),b] in A ) ) ) ) ) holds
b1 = b2
end;
:: deftheorem Def1 defines Free ZF_MODEL:def 1 :
for
H being
ZF-formula for
b2 being
set holds
(
b2 = Free H iff ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),{x,y}] in A &
[(x 'in' y),{x,y}] in A ) ) &
[H,b2] in A & ( for
H' being
ZF-formula for
a being
set st
[H',a] in A holds
( (
H' is_equality implies
a = {(Var1 H'),(Var2 H')} ) & (
H' is_membership implies
a = {(Var1 H'),(Var2 H')} ) & (
H' is
negative implies ex
b being
set st
(
a = b &
[(the_argument_of H'),b] in A ) ) & (
H' is
conjunctive implies ex
b,
c being
set st
(
a = union {b,c} &
[(the_left_argument_of H'),b] in A &
[(the_right_argument_of H'),c] in A ) ) & (
H' is
universal implies ex
b being
set st
(
a = (union {b}) \ {(bound_in H')} &
[(the_scope_of H'),b] in A ) ) ) ) ) );
deffunc H6( ZF-formula) -> set = Free a1;
Lemma111:
for H being ZF-formula
for a1 being set holds
( a1 = H6(H) iff ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),H1(x,y)] in A & [(x 'in' y),H2(x,y)] in A ) ) & [H,a1] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is_equality implies a = H1( Var1 H', Var2 H') ) & ( H' is_membership implies a = H2( Var1 H', Var2 H') ) & ( H' is negative implies ex b being set st
( a = H3(b) & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = H4(b,c) & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = H5( bound_in H',b) & [(the_scope_of H'),b] in A ) ) ) ) ) )
by ;
theorem Th1: :: ZF_MODEL:1
:: deftheorem Def2 defines VAL ZF_MODEL:def 2 :
definition
let H be
ZF-formula;
let E be non
empty set ;
deffunc H7(
Variable,
Variable)
-> set =
{ v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . a1 = f . a2 } ;
deffunc H8(
Variable,
Variable)
-> set =
{ v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . a1 in f . a2 } ;
deffunc H9(
set )
-> Element of
bool (VAL E) =
(VAL E) \ (union {a1});
deffunc H10(
set ,
set )
-> set =
(union {a1}) /\ (union {a2});
deffunc H11(
Variable,
set )
-> set =
{ v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = a2 & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
a1 = y ) holds
g in X ) ) } ;
func St c1,
c2 -> set means :
Def3:
:: ZF_MODEL:def 3
ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),{ v1 where v1 is Element of VAL E : for f being Function of VAR ,E st f = v1 holds
f . x = f . y } ] in A &
[(x 'in' y),{ v2 where v2 is Element of VAL E : for f being Function of VAR ,E st f = v2 holds
f . x in f . y } ] in A ) ) &
[H,it] in A & ( for
H' being
ZF-formula for
a being
set st
[H',a] in A holds
( (
H' is_equality implies
a = { v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . (Var1 H') = f . (Var2 H') } ) & (
H' is_membership implies
a = { v4 where v4 is Element of VAL E : for f being Function of VAR ,E st f = v4 holds
f . (Var1 H') in f . (Var2 H') } ) & (
H' is
negative implies ex
b being
set st
(
a = (VAL E) \ (union {b}) &
[(the_argument_of H'),b] in A ) ) & (
H' is
conjunctive implies ex
b,
c being
set st
(
a = (union {b}) /\ (union {c}) &
[(the_left_argument_of H'),b] in A &
[(the_right_argument_of H'),c] in A ) ) & (
H' is
universal implies ex
b being
set st
(
a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
bound_in H' = y ) holds
g in X ) ) } &
[(the_scope_of H'),b] in A ) ) ) ) );
existence
ex b1, A being set st
( ( for x, y being Variable holds
( [(x '=' y),{ v1 where v1 is Element of VAL E : for f being Function of VAR ,E st f = v1 holds
f . x = f . y } ] in A & [(x 'in' y),{ v2 where v2 is Element of VAL E : for f being Function of VAR ,E st f = v2 holds
f . x in f . y } ] in A ) ) & [H,b1] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . (Var1 H') = f . (Var2 H') } ) & ( H' is_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR ,E st f = v4 holds
f . (Var1 H') in f . (Var2 H') } ) & ( H' is negative implies ex b being set st
( a = (VAL E) \ (union {b}) & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = (union {b}) /\ (union {c}) & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
bound_in H' = y ) holds
g in X ) ) } & [(the_scope_of H'),b] in A ) ) ) ) )
uniqueness
for b1, b2 being set st ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{ v1 where v1 is Element of VAL E : for f being Function of VAR ,E st f = v1 holds
f . x = f . y } ] in A & [(x 'in' y),{ v2 where v2 is Element of VAL E : for f being Function of VAR ,E st f = v2 holds
f . x in f . y } ] in A ) ) & [H,b1] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . (Var1 H') = f . (Var2 H') } ) & ( H' is_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR ,E st f = v4 holds
f . (Var1 H') in f . (Var2 H') } ) & ( H' is negative implies ex b being set st
( a = (VAL E) \ (union {b}) & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = (union {b}) /\ (union {c}) & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
bound_in H' = y ) holds
g in X ) ) } & [(the_scope_of H'),b] in A ) ) ) ) ) & ex A being set st
( ( for x, y being Variable holds
( [(x '=' y),{ v1 where v1 is Element of VAL E : for f being Function of VAR ,E st f = v1 holds
f . x = f . y } ] in A & [(x 'in' y),{ v2 where v2 is Element of VAL E : for f being Function of VAR ,E st f = v2 holds
f . x in f . y } ] in A ) ) & [H,b2] in A & ( for H' being ZF-formula
for a being set st [H',a] in A holds
( ( H' is_equality implies a = { v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . (Var1 H') = f . (Var2 H') } ) & ( H' is_membership implies a = { v4 where v4 is Element of VAL E : for f being Function of VAR ,E st f = v4 holds
f . (Var1 H') in f . (Var2 H') } ) & ( H' is negative implies ex b being set st
( a = (VAL E) \ (union {b}) & [(the_argument_of H'),b] in A ) ) & ( H' is conjunctive implies ex b, c being set st
( a = (union {b}) /\ (union {c}) & [(the_left_argument_of H'),b] in A & [(the_right_argument_of H'),c] in A ) ) & ( H' is universal implies ex b being set st
( a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
bound_in H' = y ) holds
g in X ) ) } & [(the_scope_of H'),b] in A ) ) ) ) ) holds
b1 = b2
end;
:: deftheorem Def3 defines St ZF_MODEL:def 3 :
for
H being
ZF-formula for
E being non
empty set for
b3 being
set holds
(
b3 = St H,
E iff ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),{ v1 where v1 is Element of VAL E : for f being Function of VAR ,E st f = v1 holds
f . x = f . y } ] in A &
[(x 'in' y),{ v2 where v2 is Element of VAL E : for f being Function of VAR ,E st f = v2 holds
f . x in f . y } ] in A ) ) &
[H,b3] in A & ( for
H' being
ZF-formula for
a being
set st
[H',a] in A holds
( (
H' is_equality implies
a = { v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . (Var1 H') = f . (Var2 H') } ) & (
H' is_membership implies
a = { v4 where v4 is Element of VAL E : for f being Function of VAR ,E st f = v4 holds
f . (Var1 H') in f . (Var2 H') } ) & (
H' is
negative implies ex
b being
set st
(
a = (VAL E) \ (union {b}) &
[(the_argument_of H'),b] in A ) ) & (
H' is
conjunctive implies ex
b,
c being
set st
(
a = (union {b}) /\ (union {c}) &
[(the_left_argument_of H'),b] in A &
[(the_right_argument_of H'),c] in A ) ) & (
H' is
universal implies ex
b being
set st
(
a = { v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = b & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
bound_in H' = y ) holds
g in X ) ) } &
[(the_scope_of H'),b] in A ) ) ) ) ) );
E126:
now
let H be
ZF-formula;
let E be non
empty set ;
deffunc H7(
Variable,
Variable)
-> set =
{ v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . a1 = f . a2 } ;
deffunc H8(
Variable,
Variable)
-> set =
{ v3 where v3 is Element of VAL E : for f being Function of VAR ,E st f = v3 holds
f . a1 in f . a2 } ;
deffunc H9(
set )
-> Element of
bool (VAL E) =
(VAL E) \ (union {a1});
deffunc H10(
set ,
set )
-> set =
(union {a1}) /\ (union {a2});
deffunc H11(
Variable,
set )
-> set =
{ v5 where v5 is Element of VAL E : for X being set
for f being Function of VAR ,E st X = a2 & f = v5 holds
( f in X & ( for g being Function of VAR ,E st ( for y being Variable st g . y <> f . y holds
a1 = y ) holds
g in X ) ) } ;
deffunc H12(
ZF-formula)
-> set =
St a1,
E;
E30:
for
H being
ZF-formula for
a being
set holds
(
a = H12(
H) iff ex
A being
set st
( ( for
x,
y being
Variable holds
(
[(x '=' y),H7(x,y)] in A &
[(x 'in' y),H8(x,y)] in A ) ) &
[H,a] in A & ( for
H' being
ZF-formula for
a being
set st
[H',a] in A holds
( (
H' is_equality implies
a = H7(
Var1 H',
Var2 H') ) & (
H' is_membership implies
a = H8(
Var1 H',
Var2 H') ) & (
H' is
negative implies ex
b being
set st
(
a = H9(
b) &
[(the_argument_of H'),b] in A ) ) & (
H' is
conjunctive implies ex
b,
c being
set st
(
a = H10(
b,
c) &
[(the_left_argument_of H'),b] in A &
[(the_right_argument_of H'),c] in A ) ) & (
H' is
universal implies ex
b being
set st
(
a = H11(
bound_in H',
b) &
[(the_scope_of H'),b] in A ) ) ) ) ) )
by ;
thus
( (
H is_equality implies
H12(
H)
= H7(
Var1 H,
Var2 H) ) & (
H is_membership implies
H12(
H)
= H8(
Var1 H,
Var2 H) ) & (
H is
negative implies
H12(
H)
= H9(
H12(
the_argument_of H)) ) & (
H is
conjunctive implies for
a,
b being
set st
a = H12(
the_left_argument_of H) &
b = H12(
the_right_argument_of H) holds
H12(
H)
= H10(
a,
b) ) & (
H is
universal implies
H12(
H)
= H11(
bound_in H,
H12(
the_scope_of H)) ) )
from ZF_MODEL:sch 3();
end;
theorem Th2: :: ZF_MODEL:2
theorem Th3: :: ZF_MODEL:3
theorem Th4: :: ZF_MODEL:4
theorem Th5: :: ZF_MODEL:5
theorem Th6: :: ZF_MODEL:6
theorem Th7: :: ZF_MODEL:7
theorem Th8: :: ZF_MODEL:8
theorem Th9: :: ZF_MODEL:9
theorem Th10: :: ZF_MODEL:10
theorem Th11: :: ZF_MODEL:11
:: deftheorem Def4 defines |= ZF_MODEL:def 4 :
theorem Th12: :: ZF_MODEL:12
theorem Th13: :: ZF_MODEL:13
theorem Th14: :: ZF_MODEL:14
theorem Th15: :: ZF_MODEL:15
theorem Th16: :: ZF_MODEL:16
theorem Th17: :: ZF_MODEL:17
theorem Th18: :: ZF_MODEL:18
theorem Th19: :: ZF_MODEL:19
theorem Th20: :: ZF_MODEL:20
theorem Th21: :: ZF_MODEL:21
theorem Th22: :: ZF_MODEL:22
theorem Th23: :: ZF_MODEL:23
:: deftheorem Def5 defines |= ZF_MODEL:def 5 :
theorem Th24: :: ZF_MODEL:24
canceled;
theorem Th25: :: ZF_MODEL:25
definition
func the_axiom_of_extensionality -> ZF-formula equals :: ZF_MODEL:def 6
All (x. 0),
(x. 1),
((All (x. 2),(((x. 2) 'in' (x. 0)) <=> ((x. 2) 'in' (x. 1)))) => ((x. 0) '=' (x. 1)));
correctness
coherence
All (x. 0),(x. 1),((All (x. 2),(((x. 2) 'in' (x. 0)) <=> ((x. 2) 'in' (x. 1)))) => ((x. 0) '=' (x. 1))) is ZF-formula;
;
func the_axiom_of_pairs -> ZF-formula equals :: ZF_MODEL:def 7
All (x. 0),
(x. 1),
(Ex (x. 2),(All (x. 3),(((x. 3) 'in' (x. 2)) <=> (((x. 3) '=' (x. 0)) 'or' ((x. 3) '=' (x. 1))))));
correctness
coherence
All (x. 0),(x. 1),(Ex (x. 2),(All (x. 3),(((x. 3) 'in' (x. 2)) <=> (((x. 3) '=' (x. 0)) 'or' ((x. 3) '=' (x. 1)))))) is ZF-formula;
;
func the_axiom_of_unions -> ZF-formula equals :: ZF_MODEL:def 8
All (x. 0),
(Ex (x. 1),(All (x. 2),(((x. 2) 'in' (x. 1)) <=> (Ex (x. 3),(((x. 2) 'in' (x. 3)) '&' ((x. 3) 'in' (x. 0)))))));
correctness
coherence
All (x. 0),(Ex (x. 1),(All (x. 2),(((x. 2) 'in' (x. 1)) <=> (Ex (x. 3),(((x. 2) 'in' (x. 3)) '&' ((x. 3) 'in' (x. 0))))))) is ZF-formula;
;
func the_axiom_of_infinity -> ZF-formula equals :: ZF_MODEL:def 9
Ex (x. 0),
(x. 1),
(((x. 1) 'in' (x. 0)) '&' (All (x. 2),(((x. 2) 'in' (x. 0)) => (Ex (x. 3),((((x. 3) 'in' (x. 0)) '&' ('not' ((x. 3) '=' (x. 2)))) '&' (All (x. 4),(((x. 4) 'in' (x. 2)) => ((x. 4) 'in' (x. 3)))))))));
correctness
coherence
Ex (x. 0),(x. 1),(((x. 1) 'in' (x. 0)) '&' (All (x. 2),(((x. 2) 'in' (x. 0)) => (Ex (x. 3),((((x. 3) 'in' (x. 0)) '&' ('not' ((x. 3) '=' (x. 2)))) '&' (All (x. 4),(((x. 4) 'in' (x. 2)) => ((x. 4) 'in' (x. 3))))))))) is ZF-formula;
;
func the_axiom_of_power_sets -> ZF-formula equals :: ZF_MODEL:def 10
All (x. 0),
(Ex (x. 1),(All (x. 2),(((x. 2) 'in' (x. 1)) <=> (All (x. 3),(((x. 3) 'in' (x. 2)) => ((x. 3) 'in' (x. 0)))))));
correctness
coherence
All (x. 0),(Ex (x. 1),(All (x. 2),(((x. 2) 'in' (x. 1)) <=> (All (x. 3),(((x. 3) 'in' (x. 2)) => ((x. 3) 'in' (x. 0))))))) is ZF-formula;
;
end;
:: deftheorem Def6 defines the_axiom_of_extensionality ZF_MODEL:def 6 :
:: deftheorem Def7 defines the_axiom_of_pairs ZF_MODEL:def 7 :
:: deftheorem Def8 defines the_axiom_of_unions ZF_MODEL:def 8 :
:: deftheorem Def9 defines the_axiom_of_infinity ZF_MODEL:def 9 :
:: deftheorem Def10 defines the_axiom_of_power_sets ZF_MODEL:def 10 :
definition
let H be
ZF-formula;
func the_axiom_of_substitution_for c1 -> ZF-formula equals :: ZF_MODEL:def 11
(All (x. 3),(Ex (x. 0),(All (x. 4),(H <=> ((x. 4) '=' (x. 0)))))) => (All (x. 1),(Ex (x. 2),(All (x. 4),(((x. 4) 'in' (x. 2)) <=> (Ex (x. 3),(((x. 3) 'in' (x. 1)) '&' H))))));
correctness
coherence
(All (x. 3),(Ex (x. 0),(All (x. 4),(H <=> ((x. 4) '=' (x. 0)))))) => (All (x. 1),(Ex (x. 2),(All (x. 4),(((x. 4) 'in' (x. 2)) <=> (Ex (x. 3),(((x. 3) 'in' (x. 1)) '&' H)))))) is ZF-formula;
;
end;
:: deftheorem Def11 defines the_axiom_of_substitution_for ZF_MODEL:def 11 :
:: deftheorem Def12 defines being_a_model_of_ZF ZF_MODEL:def 12 :