Article ZFMODEL1, MML version 4.99.1005
:: ZFMODEL1:th 1
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_extensionality;
:: ZFMODEL1:th 2
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_pairs
iff
for b2, b3 being Element of b1 holds
{b2,b3} in b1;
:: ZFMODEL1:th 3
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_pairs
iff
for b2, b3 being set
st b2 in b1 & b3 in b1
holds {b2,b3} in b1;
:: ZFMODEL1:th 4
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_unions
iff
for b2 being Element of b1 holds
union b2 in b1;
:: ZFMODEL1:th 5
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_unions
iff
for b2 being set
st b2 in b1
holds union b2 in b1;
:: ZFMODEL1:th 6
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_infinity
iff
ex b2 being Element of b1 st
b2 <> {} &
(for b3 being Element of b1
st b3 in b2
holds ex b4 being Element of b1 st
b3 c< b4 & b4 in b2);
:: ZFMODEL1:th 7
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_infinity
iff
ex b2 being set st
b2 in b1 &
b2 <> {} &
(for b3 being set
st b3 in b2
holds ex b4 being set st
b3 c< b4 & b4 in b2);
:: ZFMODEL1:th 8
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_power_sets
iff
for b2 being Element of b1 holds
b1 /\ bool b2 in b1;
:: ZFMODEL1:th 9
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds b1 |= the_axiom_of_power_sets
iff
for b2 being set
st b2 in b1
holds b1 /\ bool b2 in b1;
:: ZFMODEL1:th 10
theorem
for b1 being Element of VAR
for b2 being ZF-formula-like FinSequence of NAT
for b3 being non empty set
for b4 being Function-like quasi_total Relation of VAR,b3
st not b1 in Free b2 & b3,b4 |= b2
holds b3,b4 |= All(b1,b2);
:: ZFMODEL1:th 11
theorem
for b1, b2 being Element of VAR
for b3 being ZF-formula-like FinSequence of NAT
for b4 being non empty set
for b5 being Function-like quasi_total Relation of VAR,b4
st {b1,b2} misses Free b3 & b4,b5 |= b3
holds b4,b5 |= All(b1,b2,b3);
:: ZFMODEL1:th 12
theorem
for b1, b2, b3 being Element of VAR
for b4 being ZF-formula-like FinSequence of NAT
for b5 being non empty set
for b6 being Function-like quasi_total Relation of VAR,b5
st {b1,b2,b3} misses Free b4 & b5,b6 |= b4
holds b5,b6 |= All(b1,b2,b3,b4);
:: ZFMODEL1:funcnot 1 => ZFMODEL1:func 1
definition
let a1 be ZF-formula-like FinSequence of NAT;
let a2 be non empty set;
let a3 be Function-like quasi_total Relation of VAR,a2;
assume not x. 0 in Free a1 &
a2,a3 |= All(x. 3,Ex(x. 0,All(x. 4,a1 <=> ((x. 4) '=' x. 0))));
func def_func'(A1,A3) -> Function-like quasi_total Relation of a2,a2 means
for b1 being Function-like quasi_total Relation of VAR,a2
st for b2 being Element of VAR
st b1 . b2 <> a3 . b2 & x. 0 <> b2 & x. 3 <> b2
holds x. 4 = b2
holds a2,b1 |= a1
iff
it . (b1 . x. 3) = b1 . x. 4;
end;
:: ZFMODEL1:def 1
theorem
for b1 being ZF-formula-like FinSequence of NAT
for b2 being non empty set
for b3 being Function-like quasi_total Relation of VAR,b2
st not x. 0 in Free b1 &
b2,b3 |= All(x. 3,Ex(x. 0,All(x. 4,b1 <=> ((x. 4) '=' x. 0))))
for b4 being Function-like quasi_total Relation of b2,b2 holds
b4 = def_func'(b1,b3)
iff
for b5 being Function-like quasi_total Relation of VAR,b2
st for b6 being Element of VAR
st b5 . b6 <> b3 . b6 & x. 0 <> b6 & x. 3 <> b6
holds x. 4 = b6
holds b2,b5 |= b1
iff
b4 . (b5 . x. 3) = b5 . x. 4;
:: ZFMODEL1:th 14
theorem
for b1 being non empty set
for b2 being ZF-formula-like FinSequence of NAT
for b3, b4 being Function-like quasi_total Relation of VAR,b1
st (for b5 being Element of VAR
st b3 . b5 <> b4 . b5
holds not b5 in Free b2) &
b1,b3 |= b2
holds b1,b4 |= b2;
:: ZFMODEL1:funcnot 2 => ZFMODEL1:func 2
definition
let a1 be ZF-formula-like FinSequence of NAT;
let a2 be non empty set;
assume Free a1 c= {x. 3,x. 4} &
a2 |= All(x. 3,Ex(x. 0,All(x. 4,a1 <=> ((x. 4) '=' x. 0))));
func def_func(A1,A2) -> Function-like quasi_total Relation of a2,a2 means
for b1 being Function-like quasi_total Relation of VAR,a2 holds
a2,b1 |= a1
iff
it . (b1 . x. 3) = b1 . x. 4;
end;
:: ZFMODEL1:def 2
theorem
for b1 being ZF-formula-like FinSequence of NAT
for b2 being non empty set
st Free b1 c= {x. 3,x. 4} &
b2 |= All(x. 3,Ex(x. 0,All(x. 4,b1 <=> ((x. 4) '=' x. 0))))
for b3 being Function-like quasi_total Relation of b2,b2 holds
b3 = def_func(b1,b2)
iff
for b4 being Function-like quasi_total Relation of VAR,b2 holds
b2,b4 |= b1
iff
b3 . (b4 . x. 3) = b4 . x. 4;
:: ZFMODEL1:prednot 1 => ZFMODEL1:pred 1
definition
let a1 be Relation-like Function-like set;
let a2 be non empty set;
pred A1 is_definable_in A2 means
ex b1 being ZF-formula-like FinSequence of NAT st
Free b1 c= {x. 3,x. 4} &
a2 |= All(x. 3,Ex(x. 0,All(x. 4,b1 <=> ((x. 4) '=' x. 0)))) &
a1 = def_func(b1,a2);
end;
:: ZFMODEL1:dfs 3
definiens
let a1 be Relation-like Function-like set;
let a2 be non empty set;
To prove
a1 is_definable_in a2
it is sufficient to prove
thus ex b1 being ZF-formula-like FinSequence of NAT st
Free b1 c= {x. 3,x. 4} &
a2 |= All(x. 3,Ex(x. 0,All(x. 4,b1 <=> ((x. 4) '=' x. 0)))) &
a1 = def_func(b1,a2);
:: ZFMODEL1:def 3
theorem
for b1 being Relation-like Function-like set
for b2 being non empty set holds
b1 is_definable_in b2
iff
ex b3 being ZF-formula-like FinSequence of NAT st
Free b3 c= {x. 3,x. 4} &
b2 |= All(x. 3,Ex(x. 0,All(x. 4,b3 <=> ((x. 4) '=' x. 0)))) &
b1 = def_func(b3,b2);
:: ZFMODEL1:prednot 2 => ZFMODEL1:pred 2
definition
let a1 be Relation-like Function-like set;
let a2 be non empty set;
pred A1 is_parametrically_definable_in A2 means
ex b1 being ZF-formula-like FinSequence of NAT st
ex b2 being Function-like quasi_total Relation of VAR,a2 st
{x. 0,x. 1,x. 2} misses Free b1 &
a2,b2 |= All(x. 3,Ex(x. 0,All(x. 4,b1 <=> ((x. 4) '=' x. 0)))) &
a1 = def_func'(b1,b2);
end;
:: ZFMODEL1:dfs 4
definiens
let a1 be Relation-like Function-like set;
let a2 be non empty set;
To prove
a1 is_parametrically_definable_in a2
it is sufficient to prove
thus ex b1 being ZF-formula-like FinSequence of NAT st
ex b2 being Function-like quasi_total Relation of VAR,a2 st
{x. 0,x. 1,x. 2} misses Free b1 &
a2,b2 |= All(x. 3,Ex(x. 0,All(x. 4,b1 <=> ((x. 4) '=' x. 0)))) &
a1 = def_func'(b1,b2);
:: ZFMODEL1:def 4
theorem
for b1 being Relation-like Function-like set
for b2 being non empty set holds
b1 is_parametrically_definable_in b2
iff
ex b3 being ZF-formula-like FinSequence of NAT st
ex b4 being Function-like quasi_total Relation of VAR,b2 st
{x. 0,x. 1,x. 2} misses Free b3 &
b2,b4 |= All(x. 3,Ex(x. 0,All(x. 4,b3 <=> ((x. 4) '=' x. 0)))) &
b1 = def_func'(b3,b4);
:: ZFMODEL1:th 18
theorem
for b1 being non empty set
for b2 being Relation-like Function-like set
st b2 is_definable_in b1
holds b2 is_parametrically_definable_in b1;
:: ZFMODEL1:th 19
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds 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
iff
for b2 being ZF-formula-like FinSequence of NAT
for b3 being Function-like quasi_total Relation of VAR,b1
st {x. 0,x. 1,x. 2} misses Free b2 &
b1,b3 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0))))
for b4 being Element of b1 holds
(def_func'(b2,b3)) .: b4 in b1;
:: ZFMODEL1:th 20
theorem
for b1 being non empty set
st b1 is epsilon-transitive
holds 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
iff
for b2 being Relation-like Function-like set
st b2 is_parametrically_definable_in b1
for b3 being set
st b3 in b1
holds b2 .: b3 in b1;
:: ZFMODEL1:th 21
theorem
for b1 being non empty set
st b1 is being_a_model_of_ZF
holds b1 is epsilon-transitive &
(for b2, b3 being Element of b1
st for b4 being Element of b1 holds
b4 in b2
iff
b4 in b3
holds b2 = b3) &
(for b2, b3 being Element of b1 holds
{b2,b3} in b1) &
(for b2 being Element of b1 holds
union b2 in b1) &
(ex b2 being Element of b1 st
b2 <> {} &
(for b3 being Element of b1
st b3 in b2
holds ex b4 being Element of b1 st
b3 c< b4 & b4 in b2)) &
(for b2 being Element of b1 holds
b1 /\ bool b2 in b1) &
(for b2 being ZF-formula-like FinSequence of NAT
for b3 being Function-like quasi_total Relation of VAR,b1
st {x. 0,x. 1,x. 2} misses Free b2 &
b1,b3 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0))))
for b4 being Element of b1 holds
(def_func'(b2,b3)) .: b4 in b1);
:: ZFMODEL1:th 22
theorem
for b1 being non empty set
st b1 is epsilon-transitive &
(for b2, b3 being Element of b1 holds
{b2,b3} in b1) &
(for b2 being Element of b1 holds
union b2 in b1) &
(ex b2 being Element of b1 st
b2 <> {} &
(for b3 being Element of b1
st b3 in b2
holds ex b4 being Element of b1 st
b3 c< b4 & b4 in b2)) &
(for b2 being Element of b1 holds
b1 /\ bool b2 in b1) &
(for b2 being ZF-formula-like FinSequence of NAT
for b3 being Function-like quasi_total Relation of VAR,b1
st {x. 0,x. 1,x. 2} misses Free b2 &
b1,b3 |= All(x. 3,Ex(x. 0,All(x. 4,b2 <=> ((x. 4) '=' x. 0))))
for b4 being Element of b1 holds
(def_func'(b2,b3)) .: b4 in b1)
holds b1 is being_a_model_of_ZF;