:: RELSET_1 semantic presentation
:: deftheorem Def1 defines Relation RELSET_1:def 1 :
theorem Th1: :: RELSET_1:1
canceled;
theorem Th2: :: RELSET_1:2
canceled;
theorem Th3: :: RELSET_1:3
canceled;
theorem Th4: :: RELSET_1:4
for b
1, b
2, b
3 being
set for b
4 being
Relation of b
2,b
3 holds
( b
1 c= b
4 implies b
1 is
Relation of b
2,b
3 )
theorem Th5: :: RELSET_1:5
canceled;
theorem Th6: :: RELSET_1:6
for b
1, b
2, b
3 being
set for b
4 being
Relation of b
1,b
2 holds
not ( b
3 in b
4 & ( for b
5, b
6 being
set holds
not ( b
3 = [b5,b6] & b
5 in b
1 & b
6 in b
2 ) ) )
theorem Th7: :: RELSET_1:7
canceled;
theorem Th8: :: RELSET_1:8
for b
1, b
2, b
3, b
4 being
set holds
( b
3 in b
1 & b
4 in b
2 implies
{[b3,b4]} is
Relation of b
1,b
2 )
theorem Th9: :: RELSET_1:9
theorem Th10: :: RELSET_1:10
theorem Th11: :: RELSET_1:11
theorem Th12: :: RELSET_1:12
theorem Th13: :: RELSET_1:13
theorem Th14: :: RELSET_1:14
theorem Th15: :: RELSET_1:15
for b
1, b
2, b
3 being
set for b
4 being
Relation of b
1,b
3 holds
( b
1 c= b
2 implies b
4 is
Relation of b
2,b
3 )
theorem Th16: :: RELSET_1:16
for b
1, b
2, b
3 being
set for b
4 being
Relation of b
3,b
1 holds
( b
1 c= b
2 implies b
4 is
Relation of b
3,b
2 )
theorem Th17: :: RELSET_1:17
for b
1, b
2, b
3, b
4 being
set for b
5 being
Relation of b
1,b
3 holds
( b
1 c= b
2 & b
3 c= b
4 implies b
5 is
Relation of b
2,b
4 )
theorem Th18: :: RELSET_1:18
canceled;
theorem Th19: :: RELSET_1:19
theorem Th20: :: RELSET_1:20
canceled;
theorem Th21: :: RELSET_1:21
canceled;
theorem Th22: :: RELSET_1:22
for b
1, b
2 being
set for b
3 being
Relation of b
2,b
1 holds
( ( for b
4 being
set holds
not ( b
4 in b
2 & ( for b
5 being
set holds
not
[b4,b5] in b
3 ) ) ) iff
dom b
3 = b
2 )
theorem Th23: :: RELSET_1:23
for b
1, b
2 being
set for b
3 being
Relation of b
1,b
2 holds
( ( for b
4 being
set holds
not ( b
4 in b
2 & ( for b
5 being
set holds
not
[b5,b4] in b
3 ) ) ) iff
rng b
3 = b
2 )
theorem Th24: :: RELSET_1:24
theorem Th25: :: RELSET_1:25
theorem Th26: :: RELSET_1:26
theorem Th27: :: RELSET_1:27
theorem Th28: :: RELSET_1:28
theorem Th29: :: RELSET_1:29
theorem Th30: :: RELSET_1:30
theorem Th31: :: RELSET_1:31
theorem Th32: :: RELSET_1:32
theorem Th33: :: RELSET_1:33
theorem Th34: :: RELSET_1:34
for b
1, b
2, b
3 being
set for b
4 being
Relation of b
2,b
1 holds
( b
2 c= b
3 implies b
4 | b
3 = b
4 )
theorem Th35: :: RELSET_1:35
theorem Th36: :: RELSET_1:36
for b
1, b
2, b
3 being
set for b
4 being
Relation of b
1,b
2 holds
( b
2 c= b
3 implies b
3 | b
4 = b
4 )
theorem Th37: :: RELSET_1:37
canceled;
theorem Th38: :: RELSET_1:38
theorem Th39: :: RELSET_1:39
theorem Th40: :: RELSET_1:40
canceled;
theorem Th41: :: RELSET_1:41
canceled;
theorem Th42: :: RELSET_1:42
canceled;
theorem Th43: :: RELSET_1:43
canceled;
theorem Th44: :: RELSET_1:44
canceled;
theorem Th45: :: RELSET_1:45
canceled;
theorem Th46: :: RELSET_1:46
theorem Th47: :: RELSET_1:47
theorem Th48: :: RELSET_1:48
theorem Th49: :: RELSET_1:49
theorem Th50: :: RELSET_1:50
theorem Th51: :: RELSET_1:51
theorem Th52: :: RELSET_1:52
theorem Th53: :: RELSET_1:53