:: ORDERS_1 semantic presentation
Lemma17:
for Y being set holds
( ex X being set st
( X <> {} & X in Y ) iff union Y <> {} )
:: deftheorem Def1 defines Choice_Function ORDERS_1:def 1 :
:: deftheorem Def2 defines BOOL ORDERS_1:def 2 :
theorem Th1: :: ORDERS_1:1
canceled;
theorem Th2: :: ORDERS_1:2
canceled;
theorem Th3: :: ORDERS_1:3
canceled;
theorem Th4: :: ORDERS_1:4
theorem Th5: :: ORDERS_1:5
theorem Th6: :: ORDERS_1:6
theorem Th7: :: ORDERS_1:7
Lemma77:
for X being set
for R being total Relation of X holds field R = X
theorem Th8: :: ORDERS_1:8
canceled;
theorem Th9: :: ORDERS_1:9
canceled;
theorem Th10: :: ORDERS_1:10
canceled;
theorem Th11: :: ORDERS_1:11
canceled;
theorem Th12: :: ORDERS_1:12
theorem Th13: :: ORDERS_1:13
theorem Th14: :: ORDERS_1:14
theorem Th15: :: ORDERS_1:15
canceled;
theorem Th16: :: ORDERS_1:16
canceled;
theorem Th17: :: ORDERS_1:17
canceled;
theorem Th18: :: ORDERS_1:18
canceled;
theorem Th19: :: ORDERS_1:19
canceled;
theorem Th20: :: ORDERS_1:20
canceled;
theorem Th21: :: ORDERS_1:21
canceled;
theorem Th22: :: ORDERS_1:22
canceled;
theorem Th23: :: ORDERS_1:23
canceled;
theorem Th24: :: ORDERS_1:24
canceled;
theorem Th25: :: ORDERS_1:25
canceled;
theorem Th26: :: ORDERS_1:26
canceled;
theorem Th27: :: ORDERS_1:27
canceled;
theorem Th28: :: ORDERS_1:28
canceled;
theorem Th29: :: ORDERS_1:29
canceled;
theorem Th30: :: ORDERS_1:30
canceled;
theorem Th31: :: ORDERS_1:31
canceled;
theorem Th32: :: ORDERS_1:32
canceled;
theorem Th33: :: ORDERS_1:33
canceled;
theorem Th34: :: ORDERS_1:34
canceled;
theorem Th35: :: ORDERS_1:35
canceled;
theorem Th36: :: ORDERS_1:36
canceled;
theorem Th37: :: ORDERS_1:37
canceled;
theorem Th38: :: ORDERS_1:38
canceled;
theorem Th39: :: ORDERS_1:39
canceled;
theorem Th40: :: ORDERS_1:40
canceled;
theorem Th41: :: ORDERS_1:41
canceled;
theorem Th42: :: ORDERS_1:42
canceled;
theorem Th43: :: ORDERS_1:43
canceled;
theorem Th44: :: ORDERS_1:44
canceled;
theorem Th45: :: ORDERS_1:45
canceled;
theorem Th46: :: ORDERS_1:46
canceled;
theorem Th47: :: ORDERS_1:47
canceled;
theorem Th48: :: ORDERS_1:48
canceled;
theorem Th49: :: ORDERS_1:49
canceled;
theorem Th50: :: ORDERS_1:50
canceled;
theorem Th51: :: ORDERS_1:51
canceled;
theorem Th52: :: ORDERS_1:52
canceled;
theorem Th53: :: ORDERS_1:53
canceled;
theorem Th54: :: ORDERS_1:54
canceled;
theorem Th55: :: ORDERS_1:55
canceled;
theorem Th56: :: ORDERS_1:56
canceled;
theorem Th57: :: ORDERS_1:57
canceled;
theorem Th58: :: ORDERS_1:58
canceled;
theorem Th59: :: ORDERS_1:59
canceled;
theorem Th60: :: ORDERS_1:60
canceled;
theorem Th61: :: ORDERS_1:61
canceled;
theorem Th62: :: ORDERS_1:62
canceled;
theorem Th63: :: ORDERS_1:63
canceled;
theorem Th64: :: ORDERS_1:64
canceled;
theorem Th65: :: ORDERS_1:65
canceled;
theorem Th66: :: ORDERS_1:66
canceled;
theorem Th67: :: ORDERS_1:67
canceled;
theorem Th68: :: ORDERS_1:68
canceled;
theorem Th69: :: ORDERS_1:69
canceled;
theorem Th70: :: ORDERS_1:70
canceled;
theorem Th71: :: ORDERS_1:71
canceled;
theorem Th72: :: ORDERS_1:72
canceled;
theorem Th73: :: ORDERS_1:73
canceled;
theorem Th74: :: ORDERS_1:74
canceled;
theorem Th75: :: ORDERS_1:75
canceled;
theorem Th76: :: ORDERS_1:76
canceled;
theorem Th77: :: ORDERS_1:77
canceled;
theorem Th78: :: ORDERS_1:78
canceled;
theorem Th79: :: ORDERS_1:79
canceled;
theorem Th80: :: ORDERS_1:80
canceled;
theorem Th81: :: ORDERS_1:81
canceled;
theorem Th82: :: ORDERS_1:82
canceled;
theorem Th83: :: ORDERS_1:83
canceled;
theorem Th84: :: ORDERS_1:84
canceled;
theorem Th85: :: ORDERS_1:85
canceled;
theorem Th86: :: ORDERS_1:86
canceled;
theorem Th87: :: ORDERS_1:87
canceled;
theorem Th88: :: ORDERS_1:88
canceled;
theorem Th89: :: ORDERS_1:89
canceled;
theorem Th90: :: ORDERS_1:90
canceled;
theorem Th91: :: ORDERS_1:91
theorem Th92: :: ORDERS_1:92
theorem Th93: :: ORDERS_1:93
theorem Th94: :: ORDERS_1:94
theorem Th95: :: ORDERS_1:95
theorem Th96: :: ORDERS_1:96
theorem Th97: :: ORDERS_1:97
theorem Th98: :: ORDERS_1:98
theorem Th99: :: ORDERS_1:99
theorem Th100: :: ORDERS_1:100
:: deftheorem Def3 defines being_quasi-order ORDERS_1:def 3 :
:: deftheorem Def4 defines being_partial-order ORDERS_1:def 4 :
:: deftheorem Def5 defines being_linear-order ORDERS_1:def 5 :
theorem Th101: :: ORDERS_1:101
canceled;
theorem Th102: :: ORDERS_1:102
canceled;
theorem Th103: :: ORDERS_1:103
canceled;
theorem Th104: :: ORDERS_1:104
theorem Th105: :: ORDERS_1:105
Lemma100:
for R being Relation st R is connected holds
R ~ is connected
theorem Th106: :: ORDERS_1:106
theorem Th107: :: ORDERS_1:107
theorem Th108: :: ORDERS_1:108
theorem Th109: :: ORDERS_1:109
theorem Th110: :: ORDERS_1:110
theorem Th111: :: ORDERS_1:111
theorem Th112: :: ORDERS_1:112
Lemma103:
for R being Relation holds R c= [:(field R),(field R):]
Lemma104:
for R being Relation
for X being set st R is reflexive & X c= field R holds
field (R |_2 X) = X
theorem Th113: :: ORDERS_1:113
theorem Th114: :: ORDERS_1:114
theorem Th115: :: ORDERS_1:115
theorem Th116: :: ORDERS_1:116
canceled;
theorem Th117: :: ORDERS_1:117
canceled;
theorem Th118: :: ORDERS_1:118
canceled;
theorem Th119: :: ORDERS_1:119
theorem Th120: :: ORDERS_1:120
:: deftheorem Def6 defines quasi_orders ORDERS_1:def 6 :
:: deftheorem Def7 defines partially_orders ORDERS_1:def 7 :
:: deftheorem Def8 defines linearly_orders ORDERS_1:def 8 :
theorem Th121: :: ORDERS_1:121
canceled;
theorem Th122: :: ORDERS_1:122
canceled;
theorem Th123: :: ORDERS_1:123
canceled;
theorem Th124: :: ORDERS_1:124
theorem Th125: :: ORDERS_1:125
theorem Th126: :: ORDERS_1:126
theorem Th127: :: ORDERS_1:127
theorem Th128: :: ORDERS_1:128
Lemma112:
for R being Relation
for X being set st R is_reflexive_in X holds
R |_2 X is reflexive
Lemma113:
for R being Relation
for X being set st R is_transitive_in X holds
R |_2 X is transitive
Lemma114:
for R being Relation
for X being set st R is_antisymmetric_in X holds
R |_2 X is antisymmetric
Lemma115:
for R being Relation
for X being set st R is_connected_in X holds
R |_2 X is connected
theorem Th129: :: ORDERS_1:129
theorem Th130: :: ORDERS_1:130
theorem Th131: :: ORDERS_1:131
theorem Th132: :: ORDERS_1:132
Lemma117:
for R being Relation
for X, Y being set st R is_connected_in X & Y c= X holds
R is_connected_in Y
theorem Th133: :: ORDERS_1:133
theorem Th134: :: ORDERS_1:134
theorem Th135: :: ORDERS_1:135
Lemma118:
for R being Relation
for X being set st R is_reflexive_in X holds
R ~ is_reflexive_in X
Lemma119:
for R being Relation
for X being set st R is_transitive_in X holds
R ~ is_transitive_in X
Lemma120:
for R being Relation
for X being set st R is_antisymmetric_in X holds
R ~ is_antisymmetric_in X
Lemma121:
for R being Relation
for X being set st R is_connected_in X holds
R ~ is_connected_in X
theorem Th136: :: ORDERS_1:136
theorem Th137: :: ORDERS_1:137
theorem Th138: :: ORDERS_1:138
theorem Th139: :: ORDERS_1:139
theorem Th140: :: ORDERS_1:140
theorem Th141: :: ORDERS_1:141
theorem Th142: :: ORDERS_1:142
theorem Th143: :: ORDERS_1:143
theorem Th144: :: ORDERS_1:144
canceled;
theorem Th145: :: ORDERS_1:145
canceled;
theorem Th146: :: ORDERS_1:146
:: deftheorem Def9 defines has_upper_Zorn_property_wrt ORDERS_1:def 9 :
:: deftheorem Def10 defines has_lower_Zorn_property_wrt ORDERS_1:def 10 :
Lemma126:
for R being Relation
for X being set holds (R |_2 X) ~ = (R ~ ) |_2 X
theorem Th147: :: ORDERS_1:147
canceled;
theorem Th148: :: ORDERS_1:148
canceled;
theorem Th149: :: ORDERS_1:149
theorem Th150: :: ORDERS_1:150
theorem Th151: :: ORDERS_1:151
theorem Th152: :: ORDERS_1:152
:: deftheorem Def11 defines is_maximal_in ORDERS_1:def 11 :
:: deftheorem Def12 defines is_minimal_in ORDERS_1:def 12 :
:: deftheorem Def13 defines is_superior_of ORDERS_1:def 13 :
:: deftheorem Def14 defines is_inferior_of ORDERS_1:def 14 :
theorem Th153: :: ORDERS_1:153
canceled;
theorem Th154: :: ORDERS_1:154
canceled;
theorem Th155: :: ORDERS_1:155
canceled;
theorem Th156: :: ORDERS_1:156
canceled;
theorem Th157: :: ORDERS_1:157
theorem Th158: :: ORDERS_1:158
theorem Th159: :: ORDERS_1:159
theorem Th160: :: ORDERS_1:160
theorem Th161: :: ORDERS_1:161
theorem Th162: :: ORDERS_1:162
theorem Th163: :: ORDERS_1:163
theorem Th164: :: ORDERS_1:164
theorem Th165: :: ORDERS_1:165
theorem Th166: :: ORDERS_1:166
Lemma135:
for R being Relation
for X, Y being set st R well_orders X & Y c= X holds
R well_orders Y
theorem Th167: :: ORDERS_1:167
canceled;
theorem Th168: :: ORDERS_1:168
canceled;
theorem Th169: :: ORDERS_1:169
canceled;
theorem Th170: :: ORDERS_1:170
canceled;
theorem Th171: :: ORDERS_1:171
canceled;
theorem Th172: :: ORDERS_1:172
canceled;
theorem Th173: :: ORDERS_1:173
theorem Th174: :: ORDERS_1:174
theorem Th175: :: ORDERS_1:175
for
X being
set st
X <> {} & ( for
Z being
set st
Z c= X &
Z is
c=-linear holds
ex
Y being
set st
(
Y in X & ( for
X1 being
set st
X1 in Z holds
X1 c= Y ) ) ) holds
ex
Y being
set st
(
Y in X & ( for
Z being
set st
Z in X &
Z <> Y holds
not
Y c= Z ) )
theorem Th176: :: ORDERS_1:176
for
X being
set st
X <> {} & ( for
Z being
set st
Z c= X &
Z is
c=-linear holds
ex
Y being
set st
(
Y in X & ( for
X1 being
set st
X1 in Z holds
Y c= X1 ) ) ) holds
ex
Y being
set st
(
Y in X & ( for
Z being
set st
Z in X &
Z <> Y holds
not
Z c= Y ) )
theorem Th177: :: ORDERS_1:177
theorem Th178: :: ORDERS_1:178
scheme :: ORDERS_1:sch 222
s222{
F1()
-> non
empty set ,
P1[
set ,
set ] } :
ex
x being
Element of
F1() st
for
y being
Element of
F1() st
x <> y holds
not
P1[
x,
y]
provided
E18:
for
x being
Element of
F1() holds
P1[
x,
x]
and E19:
for
x,
y being
Element of
F1() st
P1[
x,
y] &
P1[
y,
x] holds
x = y
and E20:
for
x,
y,
z being
Element of
F1() st
P1[
x,
y] &
P1[
y,
z] holds
P1[
x,
z]
and E24:
for
X being
set st
X c= F1() & ( for
x,
y being
Element of
F1() st
x in X &
y in X &
P1[
x,
y] holds
P1[
y,
x] ) holds
ex
y being
Element of
F1() st
for
x being
Element of
F1() st
x in X holds
P1[
x,
y]
scheme :: ORDERS_1:sch 223
s223{
F1()
-> non
empty set ,
P1[
set ,
set ] } :
ex
x being
Element of
F1() st
for
y being
Element of
F1() st
x <> y holds
not
P1[
y,
x]
provided
E18:
for
x being
Element of
F1() holds
P1[
x,
x]
and E19:
for
x,
y being
Element of
F1() st
P1[
x,
y] &
P1[
y,
x] holds
x = y
and E20:
for
x,
y,
z being
Element of
F1() st
P1[
x,
y] &
P1[
y,
z] holds
P1[
x,
z]
and E24:
for
X being
set st
X c= F1() & ( for
x,
y being
Element of
F1() st
x in X &
y in X &
P1[
x,
y] holds
P1[
y,
x] ) holds
ex
y being
Element of
F1() st
for
x being
Element of
F1() st
x in X holds
P1[
y,
x]
theorem Th179: :: ORDERS_1:179
theorem Th180: :: ORDERS_1:180
theorem Th181: :: ORDERS_1:181
theorem Th182: :: ORDERS_1:182
theorem Th183: :: ORDERS_1:183
theorem Th184: :: ORDERS_1:184
theorem Th185: :: ORDERS_1:185
theorem Th186: :: ORDERS_1:186
theorem Th187: :: ORDERS_1:187
theorem Th188: :: ORDERS_1:188
theorem Th189: :: ORDERS_1:189
theorem Th190: :: ORDERS_1:190
theorem Th191: :: ORDERS_1:191
theorem Th192: :: ORDERS_1:192
theorem Th193: :: ORDERS_1:193
theorem Th194: :: ORDERS_1:194
theorem Th195: :: ORDERS_1:195
theorem Th196: :: ORDERS_1:196
theorem Th197: :: ORDERS_1:197