Article BVFUNC11, MML version 4.99.1005
:: BVFUNC11:th 1
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being a_partition of b1
st b3 is_finer_than b4
holds EqClass(b2,b3) c= EqClass(b2,b4);
:: BVFUNC11:th 2
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being a_partition of b1 holds
EqClass(b2,b3) c= EqClass(b2,b3 '\/' b4);
:: BVFUNC11:th 3
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being a_partition of b1 holds
EqClass(b2,b3 '/\' b4) c= EqClass(b2,b3);
:: BVFUNC11:th 4
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being a_partition of b1 holds
EqClass(b2,b3) c= EqClass(b2,%O b1) & EqClass(b2,SmallestPartition b1) c= EqClass(b2,b3);
:: BVFUNC11:th 5
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4} & b3 <> b4
for b5, b6 being set
st b5 in b3 & b6 in b4
holds b5 meets b6;
:: BVFUNC11:th 6
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4} & b3 <> b4
holds '/\' b2 = b3 '/\' b4;
:: BVFUNC11:th 7
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being a_partition of b1
st b2 = {b3,b4} & b3 <> b4
holds CompF(b3,b2) = b4;
:: BVFUNC11:th 8
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1
st b2 '<' b3
holds All(b2,b5,b4) '<' Ex(b3,b5,b4);
:: BVFUNC11:th 11
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(All(b2,b4,b3),b5,b3) '<' Ex(All(b2,b5,b3),b4,b3);
:: BVFUNC11:th 12
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(All(b2,b4,b3),b5,b3) '<' Ex(Ex(b2,b5,b3),b4,b3);
:: BVFUNC11:th 13
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(All(b2,b4,b3),b5,b3) '<' All(Ex(b2,b5,b3),b4,b3);
:: BVFUNC11:th 14
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(Ex(b2,b4,b3),b5,b3) '<' Ex(Ex(b2,b5,b3),b4,b3);
:: BVFUNC11:th 15
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(All(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC11:th 16
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex('not' All(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC11:th 17
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(All(b2,b4,b3),b5,b3) = Ex('not' All(b2,b5,b3),b4,b3);
:: BVFUNC11:th 18
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All('not' All(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC11:th 19
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(All(b2,b4,b3),b5,b3) = Ex(Ex('not' b2,b5,b3),b4,b3);
:: BVFUNC11:th 20
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds 'not' All(All(b2,b4,b3),b5,b3) '<' Ex(Ex('not' b2,b4,b3),b5,b3);
:: BVFUNC11:th 21
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' All(Ex(b2,b4,b3),b5,b3) = Ex(All('not' b2,b4,b3),b5,b3);
:: BVFUNC11:th 22
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' Ex(All(b2,b4,b3),b5,b3) = All(Ex('not' b2,b4,b3),b5,b3);
:: BVFUNC11:th 23
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
'not' All(All(b2,b4,b3),b5,b3) = Ex(Ex('not' b2,b4,b3),b5,b3);
:: BVFUNC11:th 24
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(All(b2,b4,b3),b5,b3) '<' Ex(Ex(b2,b5,b3),b4,b3);
:: BVFUNC11:th 25
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(All(b2,b4,b3),b5,b3) '<' All(Ex(b2,b4,b3),b5,b3);
:: BVFUNC11:th 26
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
All(All(b2,b4,b3),b5,b3) '<' Ex(Ex(b2,b4,b3),b5,b3);
:: BVFUNC11:th 27
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1 holds
Ex(All(b2,b4,b3),b5,b3) '<' Ex(Ex(b2,b4,b3),b5,b3);