:: ZFMISC_1 semantic presentation
Lemma1:
for b1 being set holds
{b1} <> {}
Lemma2:
for b1, b2 being set holds
( {b1} c= b2 iff b1 in b2 )
Lemma3:
for b1, b2, b3 being set holds
( b1 c= b2 & not b3 in b1 implies b1 c= b2 \ {b3} )
Lemma4:
for b1, b2 being set holds
( b1 c= {b2} iff ( b1 = {} or b1 = {b2} ) )
:: deftheorem Def1 defines bool ZFMISC_1:def 1 :
for b
1 being
set for b
2 being
set holds
( b
2 = bool b
1 iff for b
3 being
set holds
( b
3 in b
2 iff b
3 c= b
1 ) );
definition
let c
1, c
2 be
set ;
defpred S
1[
set ] means ex b
1, b
2 being
set st
( b
1 in c
1 & b
2 in c
2 & a
1 = [b1,b2] );
func [:c1,c2:] -> set means :
Def2:
:: ZFMISC_1:def 2
for b
1 being
set holds
( b
1 in a
3 iff ex b
2, b
3 being
set st
( b
2 in a
1 & b
3 in a
2 & b
1 = [b2,b3] ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being set st
( b3 in c1 & b4 in c2 & b2 = [b3,b4] ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = [b4,b5] ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being set st
( b4 in c1 & b5 in c2 & b3 = [b4,b5] ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines [: ZFMISC_1:def 2 :
for b
1, b
2 being
set for b
3 being
set holds
( b
3 = [:b1,b2:] iff for b
4 being
set holds
( b
4 in b
3 iff ex b
5, b
6 being
set st
( b
5 in b
1 & b
6 in b
2 & b
4 = [b5,b6] ) ) );
:: deftheorem Def3 defines [: ZFMISC_1:def 3 :
definition
let c
1, c
2, c
3, c
4 be
set ;
func [:c1,c2,c3,c4:] -> set equals :: ZFMISC_1:def 4
[:[:a1,a2,a3:],a4:];
coherence
[:[:c1,c2,c3:],c4:] is set
;
end;
:: 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
for b
1, b
2 being
set holds
(
{b1} c= {b2} implies b
1 = b
2 )
theorem Th7: :: ZFMISC_1:7
canceled;
theorem Th8: :: ZFMISC_1:8
for b
1, b
2, b
3 being
set holds
(
{b1} = {b2,b3} implies b
1 = b
2 )
theorem Th9: :: ZFMISC_1:9
for b
1, b
2, b
3 being
set holds
(
{b1} = {b2,b3} implies b
2 = b
3 )
theorem Th10: :: ZFMISC_1:10
for b
1, b
2, b
3, b
4 being
set holds
not (
{b1,b2} = {b3,b4} & not b
1 = b
3 & not b
1 = b
4 )
theorem Th11: :: ZFMISC_1:11
canceled;
theorem Th12: :: ZFMISC_1:12
Lemma12:
for b1, b2 being set holds
( {b1} \/ b2 c= b2 implies b1 in b2 )
theorem Th13: :: ZFMISC_1:13
for b
1, b
2 being
set holds
(
{b1} \/ {b2} = {b1} implies b
1 = b
2 )
Lemma13:
for b1, b2 being set holds
( b1 in b2 implies {b1} \/ b2 = b2 )
theorem Th14: :: ZFMISC_1:14
Lemma14:
for b1, b2 being set holds
not ( {b1} misses b2 & b1 in b2 )
theorem Th15: :: ZFMISC_1:15
canceled;
theorem Th16: :: ZFMISC_1:16
Lemma15:
for b1, b2 being set holds
( not b1 in b2 implies {b1} misses b2 )
theorem Th17: :: ZFMISC_1:17
Lemma17:
for b1, b2 being set holds
( b1 /\ {b2} = {b2} implies b2 in b1 )
theorem Th18: :: ZFMISC_1:18
for b
1, b
2 being
set holds
(
{b1} /\ {b2} = {b1} implies b
1 = b
2 )
Lemma18:
for b1, b2 being set holds
( b1 in b2 implies b2 /\ {b1} = {b1} )
theorem Th19: :: ZFMISC_1:19
Lemma19:
for b1, b2 being set holds
( {b1} \ b2 = {b1} iff not b1 in b2 )
theorem Th20: :: ZFMISC_1:20
Lemma20:
for b1, b2 being set holds
( {b1} \ b2 = {} iff b1 in b2 )
theorem Th21: :: ZFMISC_1:21
for b
1, b
2 being
set holds
(
{b1} \ {b2} = {} implies b
1 = b
2 )
theorem Th22: :: ZFMISC_1:22
Lemma21:
for b1, b2, b3 being set holds
( {b1,b2} \ b3 = {b1} iff ( not b1 in b3 & ( b2 in b3 or b1 = b2 ) ) )
theorem Th23: :: ZFMISC_1:23
for b
1, b
2 being
set holds
( b
1 <> b
2 implies
{b1,b2} \ {b2} = {b1} )
theorem Th24: :: ZFMISC_1:24
for b
1, b
2 being
set holds
(
{b1} c= {b2} implies b
1 = b
2 )
by Th6;
theorem Th25: :: ZFMISC_1:25
for b
1, b
2, b
3 being
set holds
not (
{b1} c= {b2,b3} & not b
1 = b
2 & not b
1 = b
3 )
theorem Th26: :: ZFMISC_1:26
for b
1, b
2, b
3 being
set holds
(
{b1,b2} c= {b3} implies b
1 = b
3 )
theorem Th27: :: ZFMISC_1:27
for b
1, b
2, b
3 being
set holds
(
{b1,b2} c= {b3} implies
{b1,b2} = {b3} )
Lemma23:
for b1, b2 being set holds
not ( b1 <> {b2} & b1 <> {} & ( for b3 being set holds
not ( b3 in b1 & b3 <> b2 ) ) )
Lemma24:
for b1, b2, b3 being set holds
( b1 c= {b2,b3} iff not ( not b1 = {} & not b1 = {b2} & not b1 = {b3} & not b1 = {b2,b3} ) )
theorem Th28: :: ZFMISC_1:28
for b
1, b
2, b
3, b
4 being
set holds
not (
{b1,b2} c= {b3,b4} & not b
1 = b
3 & not b
1 = b
4 )
theorem Th29: :: ZFMISC_1:29
theorem Th30: :: ZFMISC_1:30
Lemma25:
for b1, b2 being set holds
( b1 in b2 implies b1 c= union b2 )
theorem Th31: :: ZFMISC_1:31
Lemma26:
for b1, b2 being set holds union {b1,b2} = b1 \/ b2
theorem Th32: :: ZFMISC_1:32
theorem Th33: :: ZFMISC_1:33
for b
1, b
2, b
3, b
4 being
set holds
(
[b1,b2] = [b3,b4] implies ( b
1 = b
3 & b
2 = b
4 ) )
Lemma28:
for b1, b2, b3, b4 being set holds
( [b1,b2] in [:b3,b4:] iff ( b1 in b3 & b2 in b4 ) )
theorem Th34: :: ZFMISC_1:34
for b
1, b
2, b
3, b
4 being
set holds
(
[b1,b2] in [:{b3},{b4}:] iff ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th35: :: ZFMISC_1:35
theorem Th36: :: ZFMISC_1:36
for b
1, b
2, b
3 being
set holds
(
[:{b1},{b2,b3}:] = {[b1,b2],[b1,b3]} &
[:{b1,b2},{b3}:] = {[b1,b3],[b2,b3]} )
theorem Th37: :: ZFMISC_1:37
theorem Th38: :: ZFMISC_1:38
for b
1, b
2, b
3 being
set holds
(
{b1,b2} c= b
3 iff ( b
1 in b
3 & b
2 in b
3 ) )
theorem Th39: :: ZFMISC_1:39
theorem Th40: :: ZFMISC_1:40
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & not b
3 in b
1 implies b
1 c= b
2 \ {b3} )
by Lemma3;
theorem Th41: :: ZFMISC_1:41
for b
1, b
2 being
set holds
not ( b
1 <> {b2} & b
1 <> {} & ( for b
3 being
set holds
not ( b
3 in b
1 & b
3 <> b
2 ) ) )
by Lemma23;
theorem Th42: :: ZFMISC_1:42
for b
1, b
2, b
3 being
set holds
( b
1 c= {b2,b3} iff not ( not b
1 = {} & not b
1 = {b2} & not b
1 = {b3} & not b
1 = {b2,b3} ) )
by Lemma24;
theorem Th43: :: ZFMISC_1:43
for b
1, b
2, b
3 being
set holds
not (
{b1} = b
2 \/ b
3 & not ( b
2 = {b1} & b
3 = {b1} ) & not ( b
2 = {} & b
3 = {b1} ) & not ( b
2 = {b1} & b
3 = {} ) )
theorem Th44: :: ZFMISC_1:44
for b
1, b
2, b
3 being
set holds
not (
{b1} = b
2 \/ b
3 & b
2 <> b
3 & not b
2 = {} & not b
3 = {} )
theorem Th45: :: ZFMISC_1:45
theorem Th46: :: ZFMISC_1:46
theorem Th47: :: ZFMISC_1:47
for b
1, b
2, b
3 being
set holds
(
{b1,b2} \/ b
3 c= b
3 implies b
1 in b
3 )
theorem Th48: :: ZFMISC_1:48
for b
1, b
2, b
3 being
set holds
( b
1 in b
2 & b
3 in b
2 implies
{b1,b3} \/ b
2 = b
2 )
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
for b
1, b
2, b
3 being
set holds
( b
1 in b
2 & b
3 in b
2 implies
{b1,b3} /\ b
2 = {b1,b3} )
theorem Th54: :: ZFMISC_1:54
theorem Th55: :: ZFMISC_1:55
for b
1, b
2, b
3 being
set holds
not (
{b1,b2} misses b
3 & b
1 in b
3 )
theorem Th56: :: ZFMISC_1:56
theorem Th57: :: ZFMISC_1:57
for b
1, b
2, b
3 being
set holds
( not b
1 in b
2 & not b
3 in b
2 implies
{b1,b3} misses b
2 )
theorem Th58: :: ZFMISC_1:58
theorem Th59: :: ZFMISC_1:59
for b
1, b
2, b
3 being
set holds
not (
{b1,b2} /\ b
3 = {b1} & b
2 in b
3 & not b
1 = b
2 )
theorem Th60: :: ZFMISC_1:60
for b
1, b
2, b
3 being
set holds
( b
1 in b
2 & not ( b
3 in b
2 & not b
1 = b
3 ) implies
{b1,b3} /\ b
2 = {b1} )
theorem Th61: :: ZFMISC_1:61
canceled;
theorem Th62: :: ZFMISC_1:62
canceled;
theorem Th63: :: ZFMISC_1:63
for b
1, b
2, b
3 being
set holds
(
{b1,b2} /\ b
3 = {b1,b2} implies b
1 in b
3 )
theorem Th64: :: ZFMISC_1:64
for b
1, b
2, b
3 being
set holds
( b
1 in b
2 \ {b3} iff ( b
1 in b
2 & b
1 <> b
3 ) )
theorem Th65: :: ZFMISC_1:65
for b
1, b
2 being
set holds
( b
1 \ {b2} = b
1 iff not b
2 in b
1 )
theorem Th66: :: ZFMISC_1:66
for b
1, b
2 being
set holds
not ( b
1 \ {b2} = {} & not b
1 = {} & not b
1 = {b2} )
theorem Th67: :: ZFMISC_1:67
theorem Th68: :: ZFMISC_1:68
theorem Th69: :: ZFMISC_1:69
theorem Th70: :: ZFMISC_1:70
for b
1, b
2, b
3 being
set holds
(
{b1,b2} \ b
3 = {b1} iff ( not b
1 in b
3 & ( b
2 in b
3 or b
1 = b
2 ) ) )
by Lemma21;
theorem Th71: :: ZFMISC_1:71
canceled;
theorem Th72: :: ZFMISC_1:72
for b
1, b
2, b
3 being
set holds
(
{b1,b2} \ b
3 = {b1,b2} iff ( not b
1 in b
3 & not b
2 in b
3 ) )
theorem Th73: :: ZFMISC_1:73
for b
1, b
2, b
3 being
set holds
(
{b1,b2} \ b
3 = {} iff ( b
1 in b
3 & b
2 in b
3 ) )
theorem Th74: :: ZFMISC_1:74
for b
1, b
2, b
3 being
set holds
not ( not
{b1,b2} \ b
3 = {} & not
{b1,b2} \ b
3 = {b1} & not
{b1,b2} \ b
3 = {b2} & not
{b1,b2} \ b
3 = {b1,b2} )
theorem Th75: :: ZFMISC_1:75
for b
1, b
2, b
3 being
set holds
( b
1 \ {b2,b3} = {} iff not ( not b
1 = {} & not b
1 = {b2} & not b
1 = {b3} & not b
1 = {b2,b3} ) )
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
for b
1, b
2 being
set holds
( ( for b
3 being
set holds
( b
3 in b
1 implies b
3 c= b
2 ) ) implies
union b
1 c= b
2 )
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
for b
1, b
2, b
3 being
set holds
not ( b
1 in [:b2,b3:] & ( for b
4, b
5 being
set holds
not
[b4,b5] = b
1 ) )
theorem Th103: :: ZFMISC_1:103
for b
1, b
2, b
3, b
4 being
set holds
not ( b
1 c= [:b2,b3:] & b
4 in b
1 & ( for b
5, b
6 being
set holds
not ( b
5 in b
2 & b
6 in b
3 & b
4 = [b5,b6] ) ) )
theorem Th104: :: ZFMISC_1:104
for b
1, b
2, b
3, b
4, b
5 being
set holds
not ( b
1 in [:b2,b3:] /\ [:b4,b5:] & ( for b
6, b
7 being
set holds
not ( b
1 = [b6,b7] & b
6 in b
2 /\ b
4 & b
7 in b
3 /\ b
5 ) ) )
theorem Th105: :: ZFMISC_1:105
theorem Th106: :: ZFMISC_1:106
theorem Th107: :: ZFMISC_1:107
theorem Th108: :: ZFMISC_1:108
for b
1, b
2, b
3, b
4 being
set holds
( ( for b
5, b
6 being
set holds
(
[b5,b6] in [:b1,b2:] iff
[b5,b6] in [:b3,b4:] ) ) implies
[:b1,b2:] = [:b3,b4:] )
theorem Th109: :: ZFMISC_1:109
for b
1, b
2, b
3, b
4 being
set holds
( b
1 c= [:b2,b3:] & ( for b
5, b
6 being
set holds
(
[b5,b6] in b
1 implies
[b5,b6] in b
4 ) ) implies b
1 c= b
4 )
theorem Th110: :: ZFMISC_1:110
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( b
1 c= [:b2,b3:] & b
4 c= [:b5,b6:] & ( for b
7, b
8 being
set holds
(
[b7,b8] in b
1 iff
[b7,b8] in b
4 ) ) implies b
1 = b
4 )
theorem Th111: :: ZFMISC_1:111
for b
1, b
2 being
set holds
( ( for b
3 being
set holds
not ( b
3 in b
1 & ( for b
4, b
5 being
set holds
not b
3 = [b4,b5] ) ) ) & ( for b
3, b
4 being
set holds
(
[b3,b4] in b
1 implies
[b3,b4] in b
2 ) ) implies b
1 c= b
2 )
theorem Th112: :: ZFMISC_1:112
for b
1, b
2 being
set holds
( ( for b
3 being
set holds
not ( b
3 in b
1 & ( for b
4, b
5 being
set holds
not b
3 = [b4,b5] ) ) ) & ( for b
3 being
set holds
not ( b
3 in b
2 & ( for b
4, b
5 being
set holds
not b
3 = [b4,b5] ) ) ) & ( for b
3, b
4 being
set holds
(
[b3,b4] in b
1 iff
[b3,b4] in b
2 ) ) implies b
1 = b
2 )
theorem Th113: :: ZFMISC_1:113
theorem Th114: :: ZFMISC_1:114
theorem Th115: :: ZFMISC_1:115
for b
1, b
2 being
set holds
(
[:b1,b1:] = [:b2,b2:] implies b
1 = b
2 )
theorem Th116: :: ZFMISC_1:116
theorem Th117: :: ZFMISC_1:117
theorem Th118: :: ZFMISC_1:118
theorem Th119: :: ZFMISC_1:119
for b
1, b
2, b
3, b
4 being
set holds
( b
1 c= b
2 & b
3 c= b
4 implies
[:b1,b3:] c= [:b2,b4:] )
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
for b
1, b
2, b
3, b
4 being
set holds
(
[b1,b2] in [:{b3},b4:] iff ( b
1 = b
3 & b
2 in b
4 ) )
theorem Th129: :: ZFMISC_1:129
for b
1, b
2, b
3, b
4 being
set holds
(
[b1,b2] in [:b3,{b4}:] iff ( b
1 in b
3 & b
2 = b
4 ) )
theorem Th130: :: ZFMISC_1:130
theorem Th131: :: ZFMISC_1:131
theorem Th132: :: ZFMISC_1:132
for b
1, b
2, b
3 being
set holds
(
[:{b1,b2},b3:] = [:{b1},b3:] \/ [:{b2},b3:] &
[:b3,{b1,b2}:] = [:b3,{b1}:] \/ [:b3,{b2}:] )
theorem Th133: :: ZFMISC_1:133
canceled;
theorem Th134: :: ZFMISC_1:134
for b
1, b
2, b
3, b
4 being
set holds
( b
1 <> {} & b
2 <> {} &
[:b1,b2:] = [:b3,b4:] implies ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th135: :: ZFMISC_1:135
theorem Th136: :: ZFMISC_1:136
for b
1 being
set holds
ex b
2 being
set st
( b
1 in b
2 & ( for b
3, b
4 being
set holds
( b
3 in b
2 & b
4 c= b
3 implies b
4 in b
2 ) ) & ( for b
3 being
set holds
( b
3 in b
2 implies
bool b
3 in b
2 ) ) & ( for b
3 being
set holds
not ( b
3 c= b
2 & not b
3,b
2 are_equipotent & not b
3 in b
2 ) ) )
theorem Th137: :: ZFMISC_1:137
theorem Th138: :: ZFMISC_1:138
theorem Th139: :: ZFMISC_1:139
theorem Th140: :: ZFMISC_1:140
for b
1, b
2 being
set holds
( b
1 in b
2 implies
(b2 \ {b1}) \/ {b1} = b
2 )
theorem Th141: :: ZFMISC_1:141
for b
1, b
2 being
set holds
( not b
1 in b
2 implies
(b2 \/ {b1}) \ {b1} = b
2 )
theorem Th142: :: ZFMISC_1:142
for b
1, b
2, b
3, b
4 being
set holds
( b
4 c= {b1,b2,b3} iff not ( not b
4 = {} & not b
4 = {b1} & not b
4 = {b2} & not b
4 = {b3} & not b
4 = {b1,b2} & not b
4 = {b2,b3} & not b
4 = {b1,b3} & not b
4 = {b1,b2,b3} ) )