Article ZF_REFLE, MML version 4.99.1005
:: ZF_REFLE:th 2
theorem
for b1 being non empty universal set holds
b1 |= the_axiom_of_pairs;
:: ZF_REFLE:th 3
theorem
for b1 being non empty universal set holds
b1 |= the_axiom_of_unions;
:: ZF_REFLE:th 4
theorem
for b1 being non empty universal set
st omega in b1
holds b1 |= the_axiom_of_infinity;
:: ZF_REFLE:th 5
theorem
for b1 being non empty universal set holds
b1 |= the_axiom_of_power_sets;
:: ZF_REFLE:th 6
theorem
for b1 being non empty universal set
for b2 being ZF-formula-like FinSequence of NAT
st {x. 0,x. 1,x. 2} misses Free b2
holds b1 |= the_axiom_of_substitution_for b2;
:: ZF_REFLE:th 7
theorem
for b1 being non empty universal set
st omega in b1
holds b1 is being_a_model_of_ZF;
:: ZF_REFLE:sch 1
scheme ZF_REFLE:sch 1
{F1 -> non empty universal set,
F2 -> non empty set}:
ex b1 being Relation-like Function-like set st
proj1 b1 = F2() &
(for b2 being Element of F2() holds
ex b3 being Ordinal of F1() st
b3 = b1 . b2 &
P1[b2, b3] &
(for b4 being Ordinal of F1()
st P1[b2, b4]
holds b3 c= b4))
provided
for b1 being Element of F2() holds
ex b2 being Ordinal of F1() st
P1[b1, b2];
:: ZF_REFLE:th 8
theorem
for b1 being non empty universal set
for b2 being set holds
b2 is Ordinal of b1
iff
b2 in On b1;
:: ZF_REFLE:sch 2
scheme ZF_REFLE:sch 2
{F1 -> non empty universal set}:
ex b1 being Ordinal-Sequence of F1() st
for b2 being Ordinal of F1() holds
P1[b2, b1 . b2]
provided
for b1, b2, b3 being Ordinal of F1()
st P1[b1, b2] & P1[b1, b3]
holds b2 = b3
and
for b1 being Ordinal of F1() holds
ex b2 being Ordinal of F1() st
P1[b1, b2];
:: ZF_REFLE:sch 3
scheme ZF_REFLE:sch 3
{F1 -> non empty universal set,
F2 -> Ordinal of F1(),
F3 -> Ordinal of F1(),
F4 -> Ordinal of F1()}:
ex b1 being Ordinal-Sequence of F1() st
b1 . 0-element_of F1() = F2() &
(for b2 being Ordinal of F1() holds
b1 . succ b2 = F3(b2, b1 . b2)) &
(for b2 being Ordinal of F1()
st b2 <> 0-element_of F1() & b2 is being_limit_ordinal
holds b1 . b2 = F4(b2, b1 | b2))
:: ZF_REFLE:sch 4
scheme ZF_REFLE:sch 4
{F1 -> non empty universal set}:
for b1 being Ordinal of F1() holds
P1[b1]
provided
P1[0-element_of F1()]
and
for b1 being Ordinal of F1()
st P1[b1]
holds P1[succ b1]
and
for b1 being Ordinal of F1()
st b1 <> 0-element_of F1() &
b1 is being_limit_ordinal &
(for b2 being Ordinal of F1()
st b2 in b1
holds P1[b2])
holds P1[b1];
:: ZF_REFLE:funcnot 1 => ZF_REFLE:func 1
definition
let a1 be Relation-like Function-like set;
let a2 be non empty universal set;
let a3 be Ordinal of a2;
func union(A1,A3) -> set equals
Union (a2 | (a1 | Rank a3));
end;
:: ZF_REFLE:def 1
theorem
for b1 being Relation-like Function-like set
for b2 being non empty universal set
for b3 being Ordinal of b2 holds
union(b1,b3) = Union (b2 | (b1 | Rank b3));
:: ZF_REFLE:th 10
theorem
for b1 being Relation-like Function-like T-Sequence-like set
for b2 being ordinal set holds
b1 | Rank b2 is Relation-like Function-like T-Sequence-like set;
:: ZF_REFLE:th 11
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
for b2 being ordinal set holds
b1 | Rank b2 is Relation-like Function-like T-Sequence-like Ordinal-yielding set;
:: ZF_REFLE:th 12
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
Union b1 is ordinal set;
:: ZF_REFLE:th 13
theorem
for b1 being set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
Union (b1 | b2) is ordinal set;
:: ZF_REFLE:th 14
theorem
for b1 being ordinal set holds
On Rank b1 = b1;
:: ZF_REFLE:th 15
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
b2 | Rank b1 = b2 | b1;
:: ZF_REFLE:funcnot 2 => ZF_REFLE:func 2
definition
let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
let a2 be non empty universal set;
let a3 be Ordinal of a2;
redefine func union(a1,a3) -> Ordinal of a2;
end;
:: ZF_REFLE:th 17
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
for b3 being Ordinal-Sequence of b1 holds
union(b3,b2) = Union (b3 | b2) &
union(b3 | b2,b2) = Union (b3 | b2);
:: ZF_REFLE:funcnot 3 => ZF_REFLE:func 3
definition
let a1 be non empty universal set;
let a2, a3 be Ordinal of a1;
redefine func a2 \/ a3 -> Ordinal of a1;
commutativity;
:: for a1 being non empty universal set
:: for a2, a3 being Ordinal of a1 holds
:: a2 \/ a3 = a3 \/ a2;
idempotence;
:: for a1 being non empty universal set
:: for a2 being Ordinal of a1 holds
:: a2 \/ a2 = a2;
end;
:: ZF_REFLE:exreg 1
registration
let a1 be non empty universal set;
cluster non empty Element of a1;
end;
:: ZF_REFLE:modenot 1
definition
let a1 be non empty universal set;
mode Subclass of a1 is non empty Element of bool a1;
end;
:: ZF_REFLE:attrnot 1 => ZF_REFLE:attr 1
definition
let a1 be non empty universal set;
let a2 be T-Sequence of a1;
attr a2 is DOMAIN-yielding means
proj1 a2 = On a1;
end;
:: ZF_REFLE:dfs 2
definiens
let a1 be non empty universal set;
let a2 be T-Sequence of a1;
To prove
a2 is DOMAIN-yielding
it is sufficient to prove
thus proj1 a2 = On a1;
:: ZF_REFLE:def 5
theorem
for b1 being non empty universal set
for b2 being T-Sequence of b1 holds
b2 is DOMAIN-yielding(b1)
iff
proj1 b2 = On b1;
:: ZF_REFLE:exreg 2
registration
let a1 be non empty universal set;
cluster Relation-like non-empty Function-like T-Sequence-like DOMAIN-yielding T-Sequence of a1;
end;
:: ZF_REFLE:modenot 2
definition
let a1 be non empty universal set;
mode DOMAIN-Sequence of a1 is non-empty DOMAIN-yielding T-Sequence of a1;
end;
:: ZF_REFLE:funcnot 4 => ZF_REFLE:func 4
definition
let a1 be non empty universal set;
let a2 be non-empty DOMAIN-yielding T-Sequence of a1;
redefine func Union a2 -> non empty Element of bool a1;
end;
:: ZF_REFLE:funcnot 5 => ZF_REFLE:func 5
definition
let a1 be non empty universal set;
let a2 be non-empty DOMAIN-yielding T-Sequence of a1;
let a3 be Ordinal of a1;
redefine func a2 . a3 -> non empty Element of a1;
end;
:: ZF_REFLE:th 23
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
for b3 being non-empty DOMAIN-yielding T-Sequence of b1 holds
b2 in proj1 b3;
:: ZF_REFLE:th 24
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
for b3 being non-empty DOMAIN-yielding T-Sequence of b1 holds
b3 . b2 c= Union b3;
:: ZF_REFLE:th 25
theorem
NAT,VAR are_equipotent;
:: ZF_REFLE:th 27
theorem
for b1 being set holds
sup b1 c= succ union On b1;
:: ZF_REFLE:th 28
theorem
for b1 being non empty universal set
for b2 being set
st b2 in b1
holds sup b2 in b1;
:: ZF_REFLE:th 29
theorem
for b1 being non empty universal set
for b2 being non-empty DOMAIN-yielding T-Sequence of b1
st omega in b1 &
(for b3, b4 being Ordinal of b1
st b3 in b4
holds b2 . b3 c= b2 . b4) &
(for b3 being Ordinal of b1
st b3 <> {} & b3 is being_limit_ordinal
holds b2 . b3 = Union (b2 | b3))
for b3 being ZF-formula-like FinSequence of NAT holds
ex b4 being Ordinal-Sequence of b1 st
b4 is increasing &
b4 is continuous &
(for b5 being Ordinal of b1
st b4 . b5 = b5 & {} <> b5
for b6 being Function-like quasi_total Relation of VAR,b2 . b5 holds
Union b2,(Union b2) ! b6 |= b3
iff
b2 . b5,b6 |= b3);