:: ZFMISC_1 semantic presentation
Lemma24:
for x being set holds {x} <> {}
Lemma25:
for x, X being set holds
( {x} c= X iff x in X )
Lemma27:
for Y, X, x being set st Y c= X & not x in Y holds
Y c= X \ {x}
Lemma28:
for Y, x being set holds
( Y c= {x} iff ( Y = {} or Y = {x} ) )
:: deftheorem Def1 defines bool ZFMISC_1:def 1 :
for
X being
set for
b2 being
set holds
(
b2 = bool X iff for
Z being
set holds
(
Z in b2 iff
Z c= X ) );
definition
let X1 be
set ;
let X2 be
set ;
defpred S1[
set ]
means ex
x,
y being
set st
(
x in X1 &
y in X2 &
a1 = [x,y] );
func [:c1,c2:] -> set means :
Def2:
:: ZFMISC_1:def 2
for
z being
set holds
(
z in it iff ex
x,
y being
set st
(
x in X1 &
y in X2 &
z = [x,y] ) );
existence
ex b1 being set st
for z being set holds
( z in b1 iff ex x, y being set st
( x in X1 & y in X2 & z = [x,y] ) )
uniqueness
for b1, b2 being set st ( for z being set holds
( z in b1 iff ex x, y being set st
( x in X1 & y in X2 & z = [x,y] ) ) ) & ( for z being set holds
( z in b2 iff ex x, y being set st
( x in X1 & y in X2 & z = [x,y] ) ) ) holds
b1 = b2
end;
:: deftheorem Def2 defines [: ZFMISC_1:def 2 :
for
X1,
X2 being
set for
b3 being
set holds
(
b3 = [:X1,X2:] iff for
z being
set holds
(
z in b3 iff ex
x,
y being
set st
(
x in X1 &
y in X2 &
z = [x,y] ) ) );
:: deftheorem Def3 defines [: ZFMISC_1:def 3 :
:: deftheorem Def4 defines [: ZFMISC_1:def 4 :
theorem Th1: :: ZFMISC_1:1
theorem Th2: :: ZFMISC_1:2
theorem Th3: :: ZFMISC_1:3
canceled;
theorem Th4: :: ZFMISC_1:4
canceled;
theorem Th5: :: ZFMISC_1:5
canceled;
theorem Th6: :: ZFMISC_1:6
theorem Th7: :: ZFMISC_1:7
canceled;
theorem Th8: :: ZFMISC_1:8
for
x,
y1,
y2 being
set st
{x} = {y1,y2} holds
x = y1
theorem Th9: :: ZFMISC_1:9
for
x,
y1,
y2 being
set st
{x} = {y1,y2} holds
y1 = y2
theorem Th10: :: ZFMISC_1:10
for
x1,
x2,
y1,
y2 being
set holds
( not
{x1,x2} = {y1,y2} or
x1 = y1 or
x1 = y2 )
theorem Th11: :: ZFMISC_1:11
canceled;
theorem Th12: :: ZFMISC_1:12
Lemma42:
for x, X being set st {x} \/ X c= X holds
x in X
theorem Th13: :: ZFMISC_1:13
Lemma43:
for x, X being set st x in X holds
{x} \/ X = X
theorem Th14: :: ZFMISC_1:14
Lemma44:
for x, X being set st {x} misses X holds
not x in X
theorem Th15: :: ZFMISC_1:15
canceled;
theorem Th16: :: ZFMISC_1:16
Lemma45:
for x, X being set st not x in X holds
{x} misses X
theorem Th17: :: ZFMISC_1:17
Lemma47:
for X, x being set st X /\ {x} = {x} holds
x in X
theorem Th18: :: ZFMISC_1:18
Lemma48:
for x, X being set st x in X holds
X /\ {x} = {x}
theorem Th19: :: ZFMISC_1:19
Lemma49:
for x, X being set holds
( {x} \ X = {x} iff not x in X )
theorem Th20: :: ZFMISC_1:20
Lemma50:
for x, X being set holds
( {x} \ X = {} iff x in X )
theorem Th21: :: ZFMISC_1:21
theorem Th22: :: ZFMISC_1:22
Lemma51:
for x, y, X being set holds
( {x,y} \ X = {x} iff ( not x in X & ( y in X or x = y ) ) )
theorem Th23: :: ZFMISC_1:23
theorem Th24: :: ZFMISC_1:24
theorem Th25: :: ZFMISC_1:25
for
z,
x,
y being
set holds
( not
{z} c= {x,y} or
z = x or
z = y )
theorem Th26: :: ZFMISC_1:26
theorem Th27: :: ZFMISC_1:27
Lemma53:
for X, x being set st X <> {x} & X <> {} holds
ex y being set st
( y in X & y <> x )
Lemma54:
for Z, x1, x2 being set holds
( Z c= {x1,x2} iff ( Z = {} or Z = {x1} or Z = {x2} or Z = {x1,x2} ) )
theorem Th28: :: ZFMISC_1:28
for
x1,
x2,
y1,
y2 being
set holds
( not
{x1,x2} c= {y1,y2} or
x1 = y1 or
x1 = y2 )
theorem Th29: :: ZFMISC_1:29
theorem Th30: :: ZFMISC_1:30
Lemma61:
for X, A being set st X in A holds
X c= union A
theorem Th31: :: ZFMISC_1:31
Lemma62:
for X, Y being set holds union {X,Y} = X \/ Y
theorem Th32: :: ZFMISC_1:32
theorem Th33: :: ZFMISC_1:33
for
x1,
x2,
y1,
y2 being
set st
[x1,x2] = [y1,y2] holds
(
x1 = y1 &
x2 = y2 )
Lemma64:
for x, y, X, Y being set holds
( [x,y] in [:X,Y:] iff ( x in X & y in Y ) )
theorem Th34: :: ZFMISC_1:34
theorem Th35: :: ZFMISC_1:35
theorem Th36: :: ZFMISC_1:36
for
x,
y,
z being
set holds
(
[:{x},{y,z}:] = {[x,y],[x,z]} &
[:{x,y},{z}:] = {[x,z],[y,z]} )
theorem Th37: :: ZFMISC_1:37
for
x,
X being
set holds
(
{x} c= X iff
x in X )
by ;
theorem Th38: :: ZFMISC_1:38
for
x1,
x2,
Z being
set holds
(
{x1,x2} c= Z iff (
x1 in Z &
x2 in Z ) )
theorem Th39: :: ZFMISC_1:39
for
Y,
x being
set holds
(
Y c= {x} iff (
Y = {} or
Y = {x} ) )
by ;
theorem Th40: :: ZFMISC_1:40
for
Y,
X,
x being
set st
Y c= X & not
x in Y holds
Y c= X \ {x} by ;
theorem Th41: :: ZFMISC_1:41
theorem Th42: :: ZFMISC_1:42
theorem Th43: :: ZFMISC_1:43
theorem Th44: :: ZFMISC_1:44
theorem Th45: :: ZFMISC_1:45
theorem Th46: :: ZFMISC_1:46
for
x,
X being
set st
x in X holds
{x} \/ X = X by ;
theorem Th47: :: ZFMISC_1:47
theorem Th48: :: ZFMISC_1:48
for
x,
Z,
y being
set st
x in Z &
y in Z holds
{x,y} \/ Z = Z
theorem Th49: :: ZFMISC_1:49
theorem Th50: :: ZFMISC_1:50
theorem Th51: :: ZFMISC_1:51
theorem Th52: :: ZFMISC_1:52
theorem Th53: :: ZFMISC_1:53
theorem Th54: :: ZFMISC_1:54
theorem Th55: :: ZFMISC_1:55
theorem Th56: :: ZFMISC_1:56
theorem Th57: :: ZFMISC_1:57
theorem Th58: :: ZFMISC_1:58
theorem Th59: :: ZFMISC_1:59
for
x,
y,
X being
set holds
( not
{x,y} /\ X = {x} or not
y in X or
x = y )
theorem Th60: :: ZFMISC_1:60
for
x,
X,
y being
set st
x in X & ( not
y in X or
x = y ) holds
{x,y} /\ X = {x}
theorem Th61: :: ZFMISC_1:61
canceled;
theorem Th62: :: ZFMISC_1:62
canceled;
theorem Th63: :: ZFMISC_1:63
theorem Th64: :: ZFMISC_1:64
for
z,
X,
x being
set holds
(
z in X \ {x} iff (
z in X &
z <> x ) )
theorem Th65: :: ZFMISC_1:65
for
X,
x being
set holds
(
X \ {x} = X iff not
x in X )
theorem Th66: :: ZFMISC_1:66
theorem Th67: :: ZFMISC_1:67
for
x,
X being
set holds
(
{x} \ X = {x} iff not
x in X )
by ;
theorem Th68: :: ZFMISC_1:68
for
x,
X being
set holds
(
{x} \ X = {} iff
x in X )
by ;
theorem Th69: :: ZFMISC_1:69
theorem Th70: :: ZFMISC_1:70
for
x,
y,
X being
set holds
(
{x,y} \ X = {x} iff ( not
x in X & (
y in X or
x = y ) ) )
by ;
theorem Th71: :: ZFMISC_1:71
canceled;
theorem Th72: :: ZFMISC_1:72
for
x,
y,
X being
set holds
(
{x,y} \ X = {x,y} iff ( not
x in X & not
y in X ) )
theorem Th73: :: ZFMISC_1:73
for
x,
y,
X being
set holds
(
{x,y} \ X = {} iff (
x in X &
y in X ) )
theorem Th74: :: ZFMISC_1:74
theorem Th75: :: ZFMISC_1:75
theorem Th76: :: ZFMISC_1:76
canceled;
theorem Th77: :: ZFMISC_1:77
canceled;
theorem Th78: :: ZFMISC_1:78
canceled;
theorem Th79: :: ZFMISC_1:79
theorem Th80: :: ZFMISC_1:80
theorem Th81: :: ZFMISC_1:81
theorem Th82: :: ZFMISC_1:82
theorem Th83: :: ZFMISC_1:83
theorem Th84: :: ZFMISC_1:84
theorem Th85: :: ZFMISC_1:85
canceled;
theorem Th86: :: ZFMISC_1:86
theorem Th87: :: ZFMISC_1:87
canceled;
theorem Th88: :: ZFMISC_1:88
canceled;
theorem Th89: :: ZFMISC_1:89
canceled;
theorem Th90: :: ZFMISC_1:90
canceled;
theorem Th91: :: ZFMISC_1:91
canceled;
theorem Th92: :: ZFMISC_1:92
theorem Th93: :: ZFMISC_1:93
theorem Th94: :: ZFMISC_1:94
theorem Th95: :: ZFMISC_1:95
theorem Th96: :: ZFMISC_1:96
theorem Th97: :: ZFMISC_1:97
theorem Th98: :: ZFMISC_1:98
theorem Th99: :: ZFMISC_1:99
theorem Th100: :: ZFMISC_1:100
theorem Th101: :: ZFMISC_1:101
theorem Th102: :: ZFMISC_1:102
theorem Th103: :: ZFMISC_1:103
for
A,
X,
Y,
z being
set st
A c= [:X,Y:] &
z in A holds
ex
x,
y being
set st
(
x in X &
y in Y &
z = [x,y] )
theorem Th104: :: ZFMISC_1:104
theorem Th105: :: ZFMISC_1:105
theorem Th106: :: ZFMISC_1:106
theorem Th107: :: ZFMISC_1:107
theorem Th108: :: ZFMISC_1:108
for
X1,
Y1,
X2,
Y2 being
set st ( for
x,
y being
set holds
(
[x,y] in [:X1,Y1:] iff
[x,y] in [:X2,Y2:] ) ) holds
[:X1,Y1:] = [:X2,Y2:]
theorem Th109: :: ZFMISC_1:109
theorem Th110: :: ZFMISC_1:110
for
A,
X1,
Y1,
B,
X2,
Y2 being
set st
A c= [:X1,Y1:] &
B c= [:X2,Y2:] & ( for
x,
y being
set holds
(
[x,y] in A iff
[x,y] in B ) ) holds
A = B
theorem Th111: :: ZFMISC_1:111
for
A,
B being
set st ( for
z being
set st
z in A holds
ex
x,
y being
set st
z = [x,y] ) & ( for
x,
y being
set st
[x,y] in A holds
[x,y] in B ) holds
A c= B
theorem Th112: :: ZFMISC_1:112
for
A,
B being
set st ( for
z being
set st
z in A holds
ex
x,
y being
set st
z = [x,y] ) & ( for
z being
set st
z in B holds
ex
x,
y being
set st
z = [x,y] ) & ( for
x,
y being
set holds
(
[x,y] in A iff
[x,y] in B ) ) holds
A = B
theorem Th113: :: ZFMISC_1:113
theorem Th114: :: ZFMISC_1:114
theorem Th115: :: ZFMISC_1:115
theorem Th116: :: ZFMISC_1:116
theorem Th117: :: ZFMISC_1:117
theorem Th118: :: ZFMISC_1:118
theorem Th119: :: ZFMISC_1:119
theorem Th120: :: ZFMISC_1:120
theorem Th121: :: ZFMISC_1:121
theorem Th122: :: ZFMISC_1:122
theorem Th123: :: ZFMISC_1:123
theorem Th124: :: ZFMISC_1:124
theorem Th125: :: ZFMISC_1:125
theorem Th126: :: ZFMISC_1:126
theorem Th127: :: ZFMISC_1:127
theorem Th128: :: ZFMISC_1:128
theorem Th129: :: ZFMISC_1:129
theorem Th130: :: ZFMISC_1:130
theorem Th131: :: ZFMISC_1:131
theorem Th132: :: ZFMISC_1:132
for
x,
y,
X being
set holds
(
[:{x,y},X:] = [:{x},X:] \/ [:{y},X:] &
[:X,{x,y}:] = [:X,{x}:] \/ [:X,{y}:] )
theorem Th133: :: ZFMISC_1:133
canceled;
theorem Th134: :: ZFMISC_1:134
theorem Th135: :: ZFMISC_1:135
theorem Th136: :: ZFMISC_1:136
theorem Th137: :: ZFMISC_1:137
theorem Th138: :: ZFMISC_1:138
theorem Th139: :: ZFMISC_1:139
theorem Th140: :: ZFMISC_1:140
theorem Th141: :: ZFMISC_1:141
theorem Th142: :: ZFMISC_1:142
for
x,
y,
z,
Z being
set holds
(
Z c= {x,y,z} iff (
Z = {} or
Z = {x} or
Z = {y} or
Z = {z} or
Z = {x,y} or
Z = {y,z} or
Z = {x,z} or
Z = {x,y,z} ) )