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;