:: BVFUNC11 semantic presentation
theorem Th1: :: BVFUNC11:1
theorem Th2: :: BVFUNC11:2
theorem Th3: :: BVFUNC11:3
theorem Th4: :: BVFUNC11:4
theorem Th5: :: BVFUNC11:5
theorem Th6: :: BVFUNC11:6
theorem Th7: :: BVFUNC11:7
theorem Th8: :: BVFUNC11:8
theorem Th9: :: BVFUNC11:9
canceled;
theorem Th10: :: BVFUNC11:10
canceled;
theorem Th11: :: BVFUNC11:11
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (All b2,b4,b3),b
5,b
3 '<' Ex (All b2,b5,b3),b
4,b
3 )
theorem Th12: :: BVFUNC11:12
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (All b2,b4,b3),b
5,b
3 '<' Ex (Ex b2,b5,b3),b
4,b
3
theorem Th13: :: BVFUNC11:13
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (All b2,b4,b3),b
5,b
3 '<' All (Ex b2,b5,b3),b
4,b
3 )
theorem Th14: :: BVFUNC11:14
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (Ex b2,b4,b3),b
5,b
3 '<' Ex (Ex b2,b5,b3),b
4,b
3
theorem Th15: :: BVFUNC11:15
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (All b2,b4,b3),b5,b3) '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3
theorem Th16: :: BVFUNC11:16
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (All b2,b4,b3)),b
5,b
3 '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th17: :: BVFUNC11:17
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (All b2,b4,b3),b5,b3) = Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th18: :: BVFUNC11:18
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (All b2,b4,b3)),b
5,b
3 '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th19: :: BVFUNC11:19
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (All b2,b4,b3),b5,b3) = Ex (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th20: :: BVFUNC11:20
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (All b2,b4,b3),b5,b3) '<' Ex (Ex ('not' b2),b4,b3),b
5,b
3 )
theorem Th21: :: BVFUNC11:21
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (All (Ex b2,b4,b3),b5,b3) = Ex (All ('not' b2),b4,b3),b
5,b
3
theorem Th22: :: BVFUNC11:22
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (All b2,b4,b3),b5,b3) = All (Ex ('not' b2),b4,b3),b
5,b
3
theorem Th23: :: BVFUNC11:23
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (All (All b2,b4,b3),b5,b3) = Ex (Ex ('not' b2),b4,b3),b
5,b
3
theorem Th24: :: BVFUNC11:24
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex (All b2,b4,b3),b
5,b
3 '<' Ex (Ex b2,b5,b3),b
4,b
3 )
theorem Th25: :: BVFUNC11:25
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (All b2,b4,b3),b
5,b
3 '<' All (Ex b2,b4,b3),b
5,b
3
theorem Th26: :: BVFUNC11:26
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (All b2,b4,b3),b
5,b
3 '<' Ex (Ex b2,b4,b3),b
5,b
3
theorem Th27: :: BVFUNC11:27
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
Ex (All b2,b4,b3),b
5,b
3 '<' Ex (Ex b2,b4,b3),b
5,b
3