:: FINSET_1 semantic presentation
:: deftheorem Def1 defines finite FINSET_1:def 1 :
Lemma2:
for b1 being set holds {b1} is finite
Lemma3:
for b1, b2 being set holds
( b1 is finite & b2 is finite implies b1 \/ b2 is finite )
registration
let c
1, c
2, c
3, c
4, c
5 be
set ;
cluster {a1,a2,a3,a4,a5} -> finite ;
coherence
{c1,c2,c3,c4,c5} is finite
end;
registration
let c
1, c
2, c
3, c
4, c
5, c
6 be
set ;
cluster {a1,a2,a3,a4,a5,a6} -> finite ;
coherence
{c1,c2,c3,c4,c5,c6} is finite
end;
registration
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7} -> finite ;
coherence
{c1,c2,c3,c4,c5,c6,c7} is finite
end;
registration
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8} -> finite ;
coherence
{c1,c2,c3,c4,c5,c6,c7,c8} is finite
end;
theorem Th1: :: FINSET_1:1
canceled;
theorem Th2: :: FINSET_1:2
canceled;
theorem Th3: :: FINSET_1:3
canceled;
theorem Th4: :: FINSET_1:4
canceled;
theorem Th5: :: FINSET_1:5
canceled;
theorem Th6: :: FINSET_1:6
canceled;
theorem Th7: :: FINSET_1:7
canceled;
theorem Th8: :: FINSET_1:8
canceled;
theorem Th9: :: FINSET_1:9
canceled;
theorem Th10: :: FINSET_1:10
canceled;
theorem Th11: :: FINSET_1:11
canceled;
theorem Th12: :: FINSET_1:12
canceled;
theorem Th13: :: FINSET_1:13
theorem Th14: :: FINSET_1:14
theorem Th15: :: FINSET_1:15
theorem Th16: :: FINSET_1:16
theorem Th17: :: FINSET_1:17
theorem Th18: :: FINSET_1:18
for b
1 being
set holds
( b
1 is
finite implies for b
2 being
Subset-Family of b
1 holds
not ( b
2 <> {} & ( for b
3 being
set holds
not ( b
3 in b
2 & ( for b
4 being
set holds
( b
4 in b
2 & b
3 c= b
4 implies b
4 = b
3 ) ) ) ) ) )
Lemma9:
for b1 being set holds
( b1 is finite & ( for b2 being set holds
( b2 in b1 implies b2 is finite ) ) implies union b1 is finite )
theorem Th19: :: FINSET_1:19
theorem Th20: :: FINSET_1:20
theorem Th21: :: FINSET_1:21
theorem Th22: :: FINSET_1:22
theorem Th23: :: FINSET_1:23
theorem Th24: :: FINSET_1:24
theorem Th25: :: FINSET_1:25
theorem Th26: :: FINSET_1:26
theorem Th27: :: FINSET_1:27
theorem Th28: :: FINSET_1:28
theorem Th29: :: FINSET_1:29
theorem Th30: :: FINSET_1:30
for b
1 being
set holds
not ( b
1 is
finite & b
1 <> {} & b
1 is
c=-linear & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3 being
set holds
( b
3 in b
1 implies b
2 c= b
3 ) ) ) ) )
theorem Th31: :: FINSET_1:31
for b
1 being
set holds
not ( b
1 is
finite & b
1 <> {} & b
1 is
c=-linear & ( for b
2 being
set holds
not ( b
2 in b
1 & ( for b
3 being
set holds
( b
3 in b
1 implies b
3 c= b
2 ) ) ) ) )