Article FINSUB_1, MML version 4.99.1005
:: FINSUB_1:attrnot 1 => FINSUB_1:attr 1
definition
let a1 be set;
attr a1 is cup-closed means
for b1, b2 being set
st b1 in a1 & b2 in a1
holds b1 \/ b2 in a1;
end;
:: FINSUB_1:dfs 1
definiens
let a1 be set;
To prove
a1 is cup-closed
it is sufficient to prove
thus for b1, b2 being set
st b1 in a1 & b2 in a1
holds b1 \/ b2 in a1;
:: FINSUB_1:def 1
theorem
for b1 being set holds
b1 is cup-closed
iff
for b2, b3 being set
st b2 in b1 & b3 in b1
holds b2 \/ b3 in b1;
:: FINSUB_1:attrnot 2 => FINSUB_1:attr 2
definition
let a1 be set;
attr a1 is cap-closed means
for b1, b2 being set
st b1 in a1 & b2 in a1
holds b1 /\ b2 in a1;
end;
:: FINSUB_1:dfs 2
definiens
let a1 be set;
To prove
a1 is cap-closed
it is sufficient to prove
thus for b1, b2 being set
st b1 in a1 & b2 in a1
holds b1 /\ b2 in a1;
:: FINSUB_1:def 2
theorem
for b1 being set holds
b1 is cap-closed
iff
for b2, b3 being set
st b2 in b1 & b3 in b1
holds b2 /\ b3 in b1;
:: FINSUB_1:attrnot 3 => FINSUB_1:attr 3
definition
let a1 be set;
attr a1 is diff-closed means
for b1, b2 being set
st b1 in a1 & b2 in a1
holds b1 \ b2 in a1;
end;
:: FINSUB_1:dfs 3
definiens
let a1 be set;
To prove
a1 is diff-closed
it is sufficient to prove
thus for b1, b2 being set
st b1 in a1 & b2 in a1
holds b1 \ b2 in a1;
:: FINSUB_1:def 3
theorem
for b1 being set holds
b1 is diff-closed
iff
for b2, b3 being set
st b2 in b1 & b3 in b1
holds b2 \ b3 in b1;
:: FINSUB_1:attrnot 4 => FINSUB_1:attr 4
definition
let a1 be set;
attr a1 is preBoolean means
a1 is cup-closed & a1 is diff-closed;
end;
:: FINSUB_1:dfs 4
definiens
let a1 be set;
To prove
a1 is preBoolean
it is sufficient to prove
thus a1 is cup-closed & a1 is diff-closed;
:: FINSUB_1:def 4
theorem
for b1 being set holds
b1 is preBoolean
iff
b1 is cup-closed & b1 is diff-closed;
:: FINSUB_1:condreg 1
registration
cluster preBoolean -> cup-closed diff-closed (set);
end;
:: FINSUB_1:condreg 2
registration
cluster cup-closed diff-closed -> preBoolean (set);
end;
:: FINSUB_1:exreg 1
registration
cluster non empty cup-closed cap-closed diff-closed set;
end;
:: FINSUB_1:th 10
theorem
for b1 being set holds
b1 is preBoolean
iff
for b2, b3 being set
st b2 in b1 & b3 in b1
holds b2 \/ b3 in b1 & b2 \ b3 in b1;
:: FINSUB_1:funcnot 1 => FINSUB_1:func 1
definition
let a1 be non empty preBoolean set;
let a2, a3 be Element of a1;
redefine func a2 \/ a3 -> Element of a1;
commutativity;
:: for a1 being non empty preBoolean set
:: for a2, a3 being Element of a1 holds
:: a2 \/ a3 = a3 \/ a2;
idempotence;
:: for a1 being non empty preBoolean set
:: for a2 being Element of a1 holds
:: a2 \/ a2 = a2;
end;
:: FINSUB_1:funcnot 2 => FINSUB_1:func 2
definition
let a1 be non empty preBoolean set;
let a2, a3 be Element of a1;
redefine func a2 \ a3 -> Element of a1;
end;
:: FINSUB_1:th 13
theorem
for b1, b2 being set
for b3 being non empty preBoolean set
st b1 is Element of b3 & b2 is Element of b3
holds b1 /\ b2 is Element of b3;
:: FINSUB_1:th 14
theorem
for b1, b2 being set
for b3 being non empty preBoolean set
st b1 is Element of b3 & b2 is Element of b3
holds b1 \+\ b2 is Element of b3;
:: FINSUB_1:th 15
theorem
for b1 being non empty set
st for b2, b3 being Element of b1 holds
b2 \+\ b3 in b1 & b2 \ b3 in b1
holds b1 is preBoolean;
:: FINSUB_1:th 16
theorem
for b1 being non empty set
st for b2, b3 being Element of b1 holds
b2 \+\ b3 in b1 & b2 /\ b3 in b1
holds b1 is preBoolean;
:: FINSUB_1:th 17
theorem
for b1 being non empty set
st for b2, b3 being Element of b1 holds
b2 \+\ b3 in b1 & b2 \/ b3 in b1
holds b1 is preBoolean;
:: FINSUB_1:funcnot 3 => FINSUB_1:func 3
definition
let a1 be non empty preBoolean set;
let a2, a3 be Element of a1;
redefine func a2 /\ a3 -> Element of a1;
commutativity;
:: for a1 being non empty preBoolean set
:: for a2, a3 being Element of a1 holds
:: a2 /\ a3 = a3 /\ a2;
idempotence;
:: for a1 being non empty preBoolean set
:: for a2 being Element of a1 holds
:: a2 /\ a2 = a2;
end;
:: FINSUB_1:funcnot 4 => FINSUB_1:func 4
definition
let a1 be non empty preBoolean set;
let a2, a3 be Element of a1;
redefine func a2 \+\ a3 -> Element of a1;
commutativity;
:: for a1 being non empty preBoolean set
:: for a2, a3 being Element of a1 holds
:: a2 \+\ a3 = a3 \+\ a2;
end;
:: FINSUB_1:th 18
theorem
for b1 being non empty preBoolean set holds
{} in b1;
:: FINSUB_1:th 20
theorem
for b1 being set holds
bool b1 is preBoolean;
:: FINSUB_1:funcreg 1
registration
let a1 be set;
cluster bool a1 -> preBoolean;
end;
:: FINSUB_1:th 21
theorem
for b1, b2 being non empty preBoolean set holds
b1 /\ b2 is not empty & b1 /\ b2 is preBoolean;
:: FINSUB_1:funcnot 5 => FINSUB_1:func 5
definition
let a1 be set;
func Fin A1 -> preBoolean set means
for b1 being set holds
b1 in it
iff
b1 c= a1 & b1 is finite;
end;
:: FINSUB_1:def 5
theorem
for b1 being set
for b2 being preBoolean set holds
b2 = Fin b1
iff
for b3 being set holds
b3 in b2
iff
b3 c= b1 & b3 is finite;
:: FINSUB_1:funcreg 2
registration
let a1 be set;
cluster Fin a1 -> non empty preBoolean;
end;
:: FINSUB_1:condreg 3
registration
let a1 be set;
cluster -> finite (Element of Fin a1);
end;
:: FINSUB_1:th 23
theorem
for b1, b2 being set
st b1 c= b2
holds Fin b1 c= Fin b2;
:: FINSUB_1:th 24
theorem
for b1, b2 being set holds
Fin (b1 /\ b2) = (Fin b1) /\ Fin b2;
:: FINSUB_1:th 25
theorem
for b1, b2 being set holds
(Fin b1) \/ Fin b2 c= Fin (b1 \/ b2);
:: FINSUB_1:th 26
theorem
for b1 being set holds
Fin b1 c= bool b1;
:: FINSUB_1:th 27
theorem
for b1 being set
st b1 is finite
holds Fin b1 = bool b1;
:: FINSUB_1:th 28
theorem
Fin {} = {{}};
:: FINSUB_1:modenot 1
definition
let a1 be set;
mode Finite_Subset of a1 is Element of Fin a1;
end;
:: FINSUB_1:th 30
theorem
for b1 being set
for b2 being Element of Fin b1 holds
b2 is finite;
:: FINSUB_1:th 32
theorem
for b1 being set
for b2 being Element of Fin b1 holds
b2 is Element of bool b1;
:: FINSUB_1:th 34
theorem
for b1 being set
for b2 being Element of bool b1
st b1 is finite
holds b2 is Element of Fin b1;