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);