:: XBOOLE_1 semantic presentation
theorem Th1: :: XBOOLE_1:1
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
2 c= b
3 implies b
1 c= b
3 )
theorem Th2: :: XBOOLE_1:2
theorem Th3: :: XBOOLE_1:3
for b
1 being
set holds
( b
1 c= {} implies b
1 = {} )
theorem Th4: :: XBOOLE_1:4
for b
1, b
2, b
3 being
set holds
(b1 \/ b2) \/ b
3 = b
1 \/ (b2 \/ b3)
theorem Th5: :: XBOOLE_1:5
for b
1, b
2, b
3 being
set holds
(b1 \/ b2) \/ b
3 = (b1 \/ b3) \/ (b2 \/ b3)
theorem Th6: :: XBOOLE_1:6
for b
1, b
2 being
set holds b
1 \/ (b1 \/ b2) = b
1 \/ b
2
theorem Th7: :: XBOOLE_1:7
for b
1, b
2 being
set holds b
1 c= b
1 \/ b
2
theorem Th8: :: XBOOLE_1:8
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
3 c= b
2 implies b
1 \/ b
3 c= b
2 )
theorem Th9: :: XBOOLE_1:9
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 \/ b
3 c= b
2 \/ b
3 )
theorem Th10: :: XBOOLE_1:10
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 c= b
3 \/ b
2 )
theorem Th11: :: XBOOLE_1:11
for b
1, b
2, b
3 being
set holds
( b
1 \/ b
2 c= b
3 implies b
1 c= b
3 )
theorem Th12: :: XBOOLE_1:12
for b
1, b
2 being
set holds
( b
1 c= b
2 implies b
1 \/ b
2 = b
2 )
theorem Th13: :: XBOOLE_1:13
for b
1, b
2, b
3, b
4 being
set holds
( b
1 c= b
2 & b
3 c= b
4 implies b
1 \/ b
3 c= b
2 \/ b
4 )
theorem Th14: :: XBOOLE_1:14
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
3 c= b
2 & ( for b
4 being
set holds
( b
1 c= b
4 & b
3 c= b
4 implies b
2 c= b
4 ) ) implies b
2 = b
1 \/ b
3 )
theorem Th15: :: XBOOLE_1:15
for b
1, b
2 being
set holds
( b
1 \/ b
2 = {} implies b
1 = {} )
theorem Th16: :: XBOOLE_1:16
for b
1, b
2, b
3 being
set holds
(b1 /\ b2) /\ b
3 = b
1 /\ (b2 /\ b3)
theorem Th17: :: XBOOLE_1:17
for b
1, b
2 being
set holds b
1 /\ b
2 c= b
1
theorem Th18: :: XBOOLE_1:18
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 /\ b
3 implies b
1 c= b
2 )
theorem Th19: :: XBOOLE_1:19
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
1 c= b
3 implies b
1 c= b
2 /\ b
3 )
theorem Th20: :: XBOOLE_1:20
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
1 c= b
3 & ( for b
4 being
set holds
( b
4 c= b
2 & b
4 c= b
3 implies b
4 c= b
1 ) ) implies b
1 = b
2 /\ b
3 )
theorem Th21: :: XBOOLE_1:21
for b
1, b
2 being
set holds b
1 /\ (b1 \/ b2) = b
1
theorem Th22: :: XBOOLE_1:22
for b
1, b
2 being
set holds b
1 \/ (b1 /\ b2) = b
1
theorem Th23: :: XBOOLE_1:23
for b
1, b
2, b
3 being
set holds b
1 /\ (b2 \/ b3) = (b1 /\ b2) \/ (b1 /\ b3)
theorem Th24: :: XBOOLE_1:24
for b
1, b
2, b
3 being
set holds b
1 \/ (b2 /\ b3) = (b1 \/ b2) /\ (b1 \/ b3)
theorem Th25: :: XBOOLE_1:25
theorem Th26: :: XBOOLE_1:26
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 /\ b
3 c= b
2 /\ b
3 )
theorem Th27: :: XBOOLE_1:27
for b
1, b
2, b
3, b
4 being
set holds
( b
1 c= b
2 & b
3 c= b
4 implies b
1 /\ b
3 c= b
2 /\ b
4 )
theorem Th28: :: XBOOLE_1:28
for b
1, b
2 being
set holds
( b
1 c= b
2 implies b
1 /\ b
2 = b
1 )
theorem Th29: :: XBOOLE_1:29
for b
1, b
2, b
3 being
set holds b
1 /\ b
2 c= b
1 \/ b
3
theorem Th30: :: XBOOLE_1:30
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 \/ (b3 /\ b2) = (b1 \/ b3) /\ b
2 )
theorem Th31: :: XBOOLE_1:31
for b
1, b
2, b
3 being
set holds
(b1 /\ b2) \/ (b1 /\ b3) c= b
2 \/ b
3
Lemma17:
for b1, b2 being set holds
( b1 \ b2 = {} iff b1 c= b2 )
theorem Th32: :: XBOOLE_1:32
for b
1, b
2 being
set holds
( b
1 \ b
2 = b
2 \ b
1 implies b
1 = b
2 )
theorem Th33: :: XBOOLE_1:33
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 \ b
3 c= b
2 \ b
3 )
theorem Th34: :: XBOOLE_1:34
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
3 \ b
2 c= b
3 \ b
1 )
Lemma20:
for b1, b2, b3 being set holds b1 \ (b2 /\ b3) = (b1 \ b2) \/ (b1 \ b3)
theorem Th35: :: XBOOLE_1:35
for b
1, b
2, b
3, b
4 being
set holds
( b
1 c= b
2 & b
3 c= b
4 implies b
1 \ b
4 c= b
2 \ b
3 )
theorem Th36: :: XBOOLE_1:36
for b
1, b
2 being
set holds b
1 \ b
2 c= b
1
theorem Th37: :: XBOOLE_1:37
theorem Th38: :: XBOOLE_1:38
for b
1, b
2 being
set holds
( b
1 c= b
2 \ b
1 implies b
1 = {} )
theorem Th39: :: XBOOLE_1:39
for b
1, b
2 being
set holds b
1 \/ (b2 \ b1) = b
1 \/ b
2
theorem Th40: :: XBOOLE_1:40
for b
1, b
2 being
set holds
(b1 \/ b2) \ b
2 = b
1 \ b
2
theorem Th41: :: XBOOLE_1:41
for b
1, b
2, b
3 being
set holds
(b1 \ b2) \ b
3 = b
1 \ (b2 \/ b3)
theorem Th42: :: XBOOLE_1:42
for b
1, b
2, b
3 being
set holds
(b1 \/ b2) \ b
3 = (b1 \ b3) \/ (b2 \ b3)
theorem Th43: :: XBOOLE_1:43
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 \/ b
3 implies b
1 \ b
2 c= b
3 )
theorem Th44: :: XBOOLE_1:44
for b
1, b
2, b
3 being
set holds
( b
1 \ b
2 c= b
3 implies b
1 c= b
2 \/ b
3 )
theorem Th45: :: XBOOLE_1:45
for b
1, b
2 being
set holds
( b
1 c= b
2 implies b
2 = b
1 \/ (b2 \ b1) )
theorem Th46: :: XBOOLE_1:46
for b
1, b
2 being
set holds b
1 \ (b1 \/ b2) = {}
theorem Th47: :: XBOOLE_1:47
for b
1, b
2 being
set holds b
1 \ (b1 /\ b2) = b
1 \ b
2
theorem Th48: :: XBOOLE_1:48
for b
1, b
2 being
set holds b
1 \ (b1 \ b2) = b
1 /\ b
2
theorem Th49: :: XBOOLE_1:49
for b
1, b
2, b
3 being
set holds b
1 /\ (b2 \ b3) = (b1 /\ b2) \ b
3
theorem Th50: :: XBOOLE_1:50
for b
1, b
2, b
3 being
set holds b
1 /\ (b2 \ b3) = (b1 /\ b2) \ (b1 /\ b3)
theorem Th51: :: XBOOLE_1:51
for b
1, b
2 being
set holds
(b1 /\ b2) \/ (b1 \ b2) = b
1
theorem Th52: :: XBOOLE_1:52
for b
1, b
2, b
3 being
set holds b
1 \ (b2 \ b3) = (b1 \ b2) \/ (b1 /\ b3)
theorem Th53: :: XBOOLE_1:53
for b
1, b
2, b
3 being
set holds b
1 \ (b2 \/ b3) = (b1 \ b2) /\ (b1 \ b3)
theorem Th54: :: XBOOLE_1:54
theorem Th55: :: XBOOLE_1:55
for b
1, b
2 being
set holds
(b1 \/ b2) \ (b1 /\ b2) = (b1 \ b2) \/ (b2 \ b1)
Lemma31:
for b1, b2, b3 being set holds
( b1 c= b2 & b2 c< b3 implies b1 c< b3 )
theorem Th56: :: XBOOLE_1:56
for b
1, b
2, b
3 being
set holds
( b
1 c< b
2 & b
2 c< b
3 implies b
1 c< b
3 )
theorem Th57: :: XBOOLE_1:57
for b
1, b
2 being
set holds
not ( b
1 c< b
2 & b
2 c< b
1 ) ;
theorem Th58: :: XBOOLE_1:58
for b
1, b
2, b
3 being
set holds
( b
1 c< b
2 & b
2 c= b
3 implies b
1 c< b
3 )
theorem Th59: :: XBOOLE_1:59
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
2 c< b
3 implies b
1 c< b
3 )
by Lemma31;
theorem Th60: :: XBOOLE_1:60
for b
1, b
2 being
set holds
not ( b
1 c= b
2 & b
2 c< b
1 )
theorem Th61: :: XBOOLE_1:61
theorem Th62: :: XBOOLE_1:62
for b
1 being
set holds
not b
1 c< {}
theorem Th63: :: XBOOLE_1:63
theorem Th64: :: XBOOLE_1:64
theorem Th65: :: XBOOLE_1:65
theorem Th66: :: XBOOLE_1:66
theorem Th67: :: XBOOLE_1:67
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
1 c= b
3 & b
2 misses b
3 implies b
1 = {} )
theorem Th68: :: XBOOLE_1:68
theorem Th69: :: XBOOLE_1:69
theorem Th70: :: XBOOLE_1:70
theorem Th71: :: XBOOLE_1:71
theorem Th72: :: XBOOLE_1:72
theorem Th73: :: XBOOLE_1:73
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 \/ b
3 & b
1 misses b
3 implies b
1 c= b
2 )
theorem Th74: :: XBOOLE_1:74
theorem Th75: :: XBOOLE_1:75
theorem Th76: :: XBOOLE_1:76
theorem Th77: :: XBOOLE_1:77
for b
1, b
2, b
3 being
set holds
not ( b
1 meets b
2 & b
1 c= b
3 & not b
1 meets b
2 /\ b
3 )
theorem Th78: :: XBOOLE_1:78
for b
1, b
2, b
3 being
set holds
( b
1 misses b
2 implies b
1 /\ (b2 \/ b3) = b
1 /\ b
3 )
theorem Th79: :: XBOOLE_1:79
theorem Th80: :: XBOOLE_1:80
theorem Th81: :: XBOOLE_1:81
theorem Th82: :: XBOOLE_1:82
theorem Th83: :: XBOOLE_1:83
for b
1, b
2 being
set holds
( b
1 misses b
2 iff b
1 \ b
2 = b
1 )
theorem Th84: :: XBOOLE_1:84
theorem Th85: :: XBOOLE_1:85
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 misses b
3 \ b
2 )
theorem Th86: :: XBOOLE_1:86
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
1 misses b
3 implies b
1 c= b
2 \ b
3 )
theorem Th87: :: XBOOLE_1:87
for b
1, b
2, b
3 being
set holds
( b
1 misses b
2 implies
(b3 \ b1) \/ b
2 = (b3 \/ b2) \ b
1 )
theorem Th88: :: XBOOLE_1:88
for b
1, b
2 being
set holds
( b
1 misses b
2 implies
(b1 \/ b2) \ b
2 = b
1 )
theorem Th89: :: XBOOLE_1:89
theorem Th90: :: XBOOLE_1:90
theorem Th91: :: XBOOLE_1:91
theorem Th92: :: XBOOLE_1:92
theorem Th93: :: XBOOLE_1:93
Lemma42:
for b1, b2 being set holds b1 /\ b2 misses b1 \+\ b2
Lemma43:
for b1, b2 being set holds b1 \+\ b2 = (b1 \/ b2) \ (b1 /\ b2)
theorem Th94: :: XBOOLE_1:94
theorem Th95: :: XBOOLE_1:95
theorem Th96: :: XBOOLE_1:96
theorem Th97: :: XBOOLE_1:97
for b
1, b
2, b
3 being
set holds
( b
1 \ b
2 c= b
3 & b
2 \ b
1 c= b
3 implies b
1 \+\ b
2 c= b
3 )
by Th8;
theorem Th98: :: XBOOLE_1:98
for b
1, b
2 being
set holds b
1 \/ b
2 = b
1 \+\ (b2 \ b1)
theorem Th99: :: XBOOLE_1:99
for b
1, b
2, b
3 being
set holds
(b1 \+\ b2) \ b
3 = (b1 \ (b2 \/ b3)) \/ (b2 \ (b1 \/ b3))
theorem Th100: :: XBOOLE_1:100
for b
1, b
2 being
set holds b
1 \ b
2 = b
1 \+\ (b1 /\ b2)
theorem Th101: :: XBOOLE_1:101
theorem Th102: :: XBOOLE_1:102
for b
1, b
2, b
3 being
set holds b
1 \ (b2 \+\ b3) = (b1 \ (b2 \/ b3)) \/ ((b1 /\ b2) /\ b3)
theorem Th103: :: XBOOLE_1:103
theorem Th104: :: XBOOLE_1:104
theorem Th105: :: XBOOLE_1:105
for b
1, b
2 being
set holds
not ( b
1 c< b
2 & not b
2 \ b
1 <> {} )
theorem Th106: :: XBOOLE_1:106
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 \ b
3 implies ( b
1 c= b
2 & b
1 misses b
3 ) )
theorem Th107: :: XBOOLE_1:107
theorem Th108: :: XBOOLE_1:108
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 /\ b
3 c= b
2 )
theorem Th109: :: XBOOLE_1:109
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 implies b
1 \ b
3 c= b
2 )
theorem Th110: :: XBOOLE_1:110
for b
1, b
2, b
3 being
set holds
( b
1 c= b
2 & b
3 c= b
2 implies b
1 \+\ b
3 c= b
2 )
theorem Th111: :: XBOOLE_1:111
for b
1, b
2, b
3 being
set holds
(b1 /\ b2) \ (b3 /\ b2) = (b1 \ b3) /\ b
2
theorem Th112: :: XBOOLE_1:112
theorem Th113: :: XBOOLE_1:113
for b
1, b
2, b
3, b
4 being
set holds
((b1 \/ b2) \/ b3) \/ b
4 = b
1 \/ ((b2 \/ b3) \/ b4)
theorem Th114: :: XBOOLE_1:114
theorem Th115: :: XBOOLE_1:115
for b
1 being
set holds
not b
1 c< {}
theorem Th116: :: XBOOLE_1:116