Article ZFREFLE1, MML version 4.99.1005
:: ZFREFLE1:prednot 1 => ZFREFLE1:pred 1
definition
let a1 be non empty set;
let a2 be Element of bool WFF;
pred A1 |= A2 means
for b1 being ZF-formula-like FinSequence of NAT
st b1 in a2
holds a1 |= b1;
end;
:: ZFREFLE1:dfs 1
definiens
let a1 be non empty set;
let a2 be Element of bool WFF;
To prove
a1 |= a2
it is sufficient to prove
thus for b1 being ZF-formula-like FinSequence of NAT
st b1 in a2
holds a1 |= b1;
:: ZFREFLE1:def 1
theorem
for b1 being non empty set
for b2 being Element of bool WFF holds
b1 |= b2
iff
for b3 being ZF-formula-like FinSequence of NAT
st b3 in b2
holds b1 |= b3;
:: ZFREFLE1:prednot 2 => ZFREFLE1:pred 2
definition
let a1, a2 be non empty set;
pred A1 <==> A2 means
for b1 being ZF-formula-like FinSequence of NAT
st Free b1 = {}
holds a1 |= b1
iff
a2 |= b1;
symmetry;
:: for a1, a2 being non empty set
:: st a1 <==> a2
:: holds a2 <==> a1;
reflexivity;
:: for a1 being non empty set holds
:: a1 <==> a1;
end;
:: ZFREFLE1:dfs 2
definiens
let a1, a2 be non empty set;
To prove
a1 <==> a2
it is sufficient to prove
thus for b1 being ZF-formula-like FinSequence of NAT
st Free b1 = {}
holds a1 |= b1
iff
a2 |= b1;
:: ZFREFLE1:def 2
theorem
for b1, b2 being non empty set holds
b1 <==> b2
iff
for b3 being ZF-formula-like FinSequence of NAT
st Free b3 = {}
holds b1 |= b3
iff
b2 |= b3;
:: ZFREFLE1:prednot 3 => ZFREFLE1:pred 3
definition
let a1, a2 be non empty set;
pred A1 is_elementary_subsystem_of A2 means
a1 c= a2 &
(for b1 being ZF-formula-like FinSequence of NAT
for b2 being Function-like quasi_total Relation of VAR,a1 holds
a1,b2 |= b1
iff
a2,a2 ! b2 |= b1);
reflexivity;
:: for a1 being non empty set holds
:: a1 is_elementary_subsystem_of a1;
end;
:: ZFREFLE1:dfs 3
definiens
let a1, a2 be non empty set;
To prove
a1 is_elementary_subsystem_of a2
it is sufficient to prove
thus a1 c= a2 &
(for b1 being ZF-formula-like FinSequence of NAT
for b2 being Function-like quasi_total Relation of VAR,a1 holds
a1,b2 |= b1
iff
a2,a2 ! b2 |= b1);
:: ZFREFLE1:def 3
theorem
for b1, b2 being non empty set holds
b1 is_elementary_subsystem_of b2
iff
b1 c= b2 &
(for b3 being ZF-formula-like FinSequence of NAT
for b4 being Function-like quasi_total Relation of VAR,b1 holds
b1,b4 |= b3
iff
b2,b2 ! b4 |= b3);
:: ZFREFLE1:funcnot 1 => ZFREFLE1:func 1
definition
func ZF-axioms -> set means
for b1 being set holds
b1 in it
iff
b1 in WFF &
(b1 <> the_axiom_of_extensionality & b1 <> the_axiom_of_pairs & b1 <> the_axiom_of_unions & b1 <> the_axiom_of_infinity & b1 <> the_axiom_of_power_sets implies ex b2 being ZF-formula-like FinSequence of NAT st
{x. 0,x. 1,x. 2} misses Free b2 &
b1 = the_axiom_of_substitution_for b2);
end;
:: ZFREFLE1:def 4
theorem
for b1 being set holds
b1 = ZF-axioms
iff
for b2 being set holds
b2 in b1
iff
b2 in WFF &
(b2 <> the_axiom_of_extensionality & b2 <> the_axiom_of_pairs & b2 <> the_axiom_of_unions & b2 <> the_axiom_of_infinity & b2 <> the_axiom_of_power_sets implies ex b3 being ZF-formula-like FinSequence of NAT st
{x. 0,x. 1,x. 2} misses Free b3 &
b2 = the_axiom_of_substitution_for b3);
:: ZFREFLE1:funcnot 2 => ZFREFLE1:func 2
definition
redefine func ZF-axioms -> Element of bool WFF;
end;
:: ZFREFLE1:th 1
theorem
for b1 being non empty set holds
b1 |= {} WFF;
:: ZFREFLE1:th 2
theorem
for b1 being non empty set
for b2, b3 being Element of bool WFF
st b2 c= b3 & b1 |= b3
holds b1 |= b2;
:: ZFREFLE1:th 3
theorem
for b1 being non empty set
for b2, b3 being Element of bool WFF
st b1 |= b2 & b1 |= b3
holds b1 |= b2 \/ b3;
:: ZFREFLE1:th 4
theorem
for b1 being non empty set
st b1 is being_a_model_of_ZF
holds b1 |= ZF-axioms;
:: ZFREFLE1:th 5
theorem
for b1 being non empty set
st b1 |= ZF-axioms & b1 is epsilon-transitive
holds b1 is being_a_model_of_ZF;
:: ZFREFLE1:th 6
theorem
for b1 being ZF-formula-like FinSequence of NAT holds
ex b2 being ZF-formula-like FinSequence of NAT st
Free b2 = {} &
(for b3 being non empty set holds
b3 |= b2
iff
b3 |= b1);
:: ZFREFLE1:th 7
theorem
for b1, b2 being non empty set holds
b1 <==> b2
iff
for b3 being ZF-formula-like FinSequence of NAT holds
b1 |= b3
iff
b2 |= b3;
:: ZFREFLE1:th 8
theorem
for b1, b2 being non empty set holds
b1 <==> b2
iff
for b3 being Element of bool WFF holds
b1 |= b3
iff
b2 |= b3;
:: ZFREFLE1:th 9
theorem
for b1, b2 being non empty set
st b1 is_elementary_subsystem_of b2
holds b1 <==> b2;
:: ZFREFLE1:th 10
theorem
for b1, b2 being non empty set
st b1 is being_a_model_of_ZF & b1 <==> b2 & b2 is epsilon-transitive
holds b2 is being_a_model_of_ZF;
:: ZFREFLE1:th 12
theorem
for b1 being Relation-like Function-like set
for b2 being non empty universal set
st proj1 b1 in b2 & proj2 b1 c= b2
holds proj2 b1 in b2;
:: ZFREFLE1:th 13
theorem
for b1, b2 being set
st (b1,b2 are_equipotent or Card b1 = Card b2)
holds bool b1,bool b2 are_equipotent & Card bool b1 = Card bool b2;
:: ZFREFLE1:th 14
theorem
for b1 being non empty universal set
for b2 being non empty set
for b3 being Function-like quasi_total Relation of b2,Funcs(On b1,On b1)
st Card b2 in Card b1
holds ex b4 being Ordinal-Sequence of b1 st
b4 is increasing &
b4 is continuous &
b4 . 0-element_of b1 = 0-element_of b1 &
(for b5 being Ordinal of b1 holds
b4 . succ b5 = sup ({b4 . b5} \/ ((uncurry b3) .: [:b2,{succ b5}:]))) &
(for b5 being Ordinal of b1
st b5 <> 0-element_of b1 & b5 is being_limit_ordinal
holds b4 . b5 = sup (b4 | b5));
:: ZFREFLE1:th 15
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st b2 is increasing
holds b1 +^ b2 is increasing;
:: ZFREFLE1:th 16
theorem
for b1, b2 being ordinal set
for b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
(b1 +^ b3) | b2 = b1 +^ (b3 | b2);
:: ZFREFLE1:th 17
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st b2 is increasing & b2 is continuous
holds b1 +^ b2 is continuous;
:: ZFREFLE1:th 19
theorem
for b1 being set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st b1 in proj2 b2
holds b1 is ordinal set;
:: ZFREFLE1:th 20
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
proj2 b1 c= sup b1;
:: ZFREFLE1:th 21
theorem
for b1, b2, b3 being ordinal set
st b1 is_cofinal_with b2 & b2 is_cofinal_with b3
holds b1 is_cofinal_with b3;
:: ZFREFLE1:th 22
theorem
for b1, b2 being ordinal set
st b1 is_cofinal_with b2
holds b2 c= b1;
:: ZFREFLE1:th 23
theorem
for b1, b2 being ordinal set
st b1 is_cofinal_with b2 & b2 is_cofinal_with b1
holds b1 = b2;
:: ZFREFLE1:th 24
theorem
for b1 being ordinal set
for b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
st proj1 b2 <> {} & proj1 b2 is being_limit_ordinal & b2 is increasing & b1 is_limes_of b2
holds b1 is_cofinal_with proj1 b2;
:: ZFREFLE1:th 25
theorem
for b1 being ordinal set holds
succ b1 is_cofinal_with 1;
:: ZFREFLE1:th 26
theorem
for b1, b2 being ordinal set
st b1 is_cofinal_with succ b2
holds ex b3 being ordinal set st
b1 = succ b3;
:: ZFREFLE1:th 27
theorem
for b1, b2 being ordinal set
st b1 is_cofinal_with b2
holds b1 is being_limit_ordinal
iff
b2 is being_limit_ordinal;
:: ZFREFLE1:th 28
theorem
for b1 being ordinal set
st b1 is_cofinal_with {}
holds b1 = {};
:: ZFREFLE1:th 29
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1 holds
not On b1 is_cofinal_with b2;
:: ZFREFLE1:th 30
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
for b3 being Ordinal-Sequence of b1
st omega in b1 & b3 is increasing & b3 is continuous
holds ex b4 being Ordinal of b1 st
b2 in b4 & b3 . b4 = b4;
:: ZFREFLE1:th 31
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
for b3 being Ordinal-Sequence of b1
st omega in b1 & b3 is increasing & b3 is continuous
holds ex b4 being Ordinal of b1 st
b2 in b4 & b3 . b4 = b4 & b4 is_cofinal_with omega;
:: ZFREFLE1:th 32
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))
holds ex b3 being Ordinal-Sequence of b1 st
b3 is increasing &
b3 is continuous &
(for b4 being Ordinal of b1
st b3 . b4 = b4 & {} <> b4
holds b2 . b4 is_elementary_subsystem_of Union b2);
:: ZFREFLE1:th 33
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1 holds
Rank b2 in b1;
:: ZFREFLE1:th 34
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
st b2 <> {}
holds Rank b2 is non empty Element of b1;
:: ZFREFLE1:th 35
theorem
for b1 being non empty universal set
st omega in b1
holds ex b2 being Ordinal-Sequence of b1 st
b2 is increasing &
b2 is continuous &
(for b3 being Ordinal of b1
for b4 being non empty set
st b2 . b3 = b3 & {} <> b3 & b4 = Rank b3
holds b4 is_elementary_subsystem_of b1);
:: ZFREFLE1:th 36
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
st omega in b1
holds ex b3 being Ordinal of b1 st
ex b4 being non empty set st
b2 in b3 & b4 = Rank b3 & b4 is_elementary_subsystem_of b1;
:: ZFREFLE1:th 37
theorem
for b1 being non empty universal set
st omega in b1
holds ex b2 being Ordinal of b1 st
ex b3 being non empty set st
b2 is_cofinal_with omega & b3 = Rank b2 & b3 is_elementary_subsystem_of b1;
:: ZFREFLE1:th 38
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))
holds ex b3 being Ordinal-Sequence of b1 st
b3 is increasing &
b3 is continuous &
(for b4 being Ordinal of b1
st b3 . b4 = b4 & {} <> b4
holds b2 . b4 <==> Union b2);
:: ZFREFLE1:th 39
theorem
for b1 being non empty universal set
st omega in b1
holds ex b2 being Ordinal-Sequence of b1 st
b2 is increasing &
b2 is continuous &
(for b3 being Ordinal of b1
for b4 being non empty set
st b2 . b3 = b3 & {} <> b3 & b4 = Rank b3
holds b4 <==> b1);
:: ZFREFLE1:th 40
theorem
for b1 being non empty universal set
for b2 being Ordinal of b1
st omega in b1
holds ex b3 being Ordinal of b1 st
ex b4 being non empty set st
b2 in b3 & b4 = Rank b3 & b4 <==> b1;
:: ZFREFLE1:th 41
theorem
for b1 being non empty universal set
st omega in b1
holds ex b2 being Ordinal of b1 st
ex b3 being non empty set st
b2 is_cofinal_with omega & b3 = Rank b2 & b3 <==> b1;
:: ZFREFLE1:th 42
theorem
for b1 being non empty universal set
st omega in b1
holds ex b2 being Ordinal of b1 st
ex b3 being non empty set st
b2 is_cofinal_with omega & b3 = Rank b2 & b3 is being_a_model_of_ZF;
:: ZFREFLE1:th 43
theorem
for b1 being set
for b2 being non empty universal set
st omega in b2 & b1 in b2
holds ex b3 being non empty set st
b1 in b3 & b3 in b2 & b3 is being_a_model_of_ZF;