Article CARD_5, MML version 4.99.1005

:: CARD_5:th 1
theorem
1 = {0} & 2 = {0,1};

:: CARD_5:th 8
theorem
for b1 being Element of NAT holds
   Seg b1 = (b1 + 1) \ {0};

:: CARD_5:th 9
theorem
for b1 being set holds
   nextcard Card b1 = nextcard b1;

:: CARD_5:th 10
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
      b1 in Union b2
   iff
      ex b3 being set st
         b3 in proj1 b2 & b1 in b2 . b3;

:: CARD_5:th 11
theorem
for b1 being ordinal set holds
   alef b1 is infinite;

:: CARD_5:th 12
theorem
for b1 being cardinal set
      st b1 is infinite
   holds ex b2 being ordinal set st
      b1 = alef b2;

:: CARD_5:th 13
theorem
for b1 being cardinal set
      st for b2 being Element of NAT holds
           b1 <> Card b2
   holds ex b2 being ordinal set st
      b1 = alef b2;

:: CARD_5:funcreg 1
registration
  let a1 be Relation-like Function-like T-Sequence-like Ordinal-yielding set;
  cluster Union a1 -> ordinal;
end;

:: CARD_5:th 14
theorem
for b1 being ordinal set
for b2 being set
      st b2 c= b1
   holds ex b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
      b3 = canonical_isomorphism_of(RelIncl order_type_of RelIncl b2,RelIncl b2) &
       b3 is increasing &
       proj1 b3 = order_type_of RelIncl b2 &
       proj2 b3 = b2;

:: CARD_5:th 15
theorem
for b1 being ordinal set
for b2 being set
      st b2 c= b1
   holds sup b2 is_cofinal_with order_type_of RelIncl b2;

:: CARD_5:th 16
theorem
for b1 being ordinal set
for b2 being set
      st b2 c= b1
   holds Card b2 = Card order_type_of RelIncl b2;

:: CARD_5:th 17
theorem
for b1 being ordinal set holds
   ex b2 being ordinal set st
      b2 c= Card b1 & b1 is_cofinal_with b2;

:: CARD_5:th 18
theorem
for b1 being ordinal set holds
   ex b2 being cardinal set st
      b2 c= Card b1 &
       b1 is_cofinal_with b2 &
       (for b3 being ordinal set
             st b1 is_cofinal_with b3
          holds b2 c= b3);

:: CARD_5:th 19
theorem
for b1, b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
      st proj2 b1 = proj2 b2 & b1 is increasing & b2 is increasing
   holds b1 = b2;

:: CARD_5:th 20
theorem
for b1 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
      st b1 is increasing
   holds b1 is one-to-one;

:: CARD_5:th 21
theorem
for b1, b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set holds
(b1 ^ b2) | proj1 b1 = b1;

:: CARD_5:th 22
theorem
for b1 being set
for b2 being cardinal set
      st b1 <> {}
   holds Card {b3 where b3 is Element of bool b1: Card b3 in b2} c= b2 *` exp(Card b1,b2);

:: CARD_5:th 23
theorem
for b1 being cardinal set holds
   b1 in exp(2,b1);

:: CARD_5:exreg 1
registration
  cluster epsilon-transitive epsilon-connected ordinal infinite cardinal set;
end;

:: CARD_5:condreg 1
registration
  cluster infinite -> non empty (set);
end;

:: CARD_5:modenot 1
definition
  mode Aleph is infinite cardinal set;
end;

:: CARD_5:funcnot 1 => CARD_5:func 1
definition
  let a1 be cardinal set;
  func cf A1 -> cardinal set means
    a1 is_cofinal_with it &
     (for b1 being cardinal set
           st a1 is_cofinal_with b1
        holds it c= b1);
end;

:: CARD_5:def 2
theorem
for b1, b2 being cardinal set holds
   b2 = cf b1
iff
   b1 is_cofinal_with b2 &
    (for b3 being cardinal set
          st b1 is_cofinal_with b3
       holds b2 c= b3);

:: CARD_5:funcnot 2 => CARD_5:func 2
definition
  let a1, a2 be cardinal set;
  func A2 -powerfunc_of A1 -> Relation-like Function-like Cardinal-yielding set means
    (for b1 being set holds
           b1 in proj1 it
        iff
           b1 in a1 & b1 is cardinal set) &
     (for b1 being cardinal set
           st b1 in a1
        holds it . b1 = exp(b1,a2));
end;

:: CARD_5:def 3
theorem
for b1, b2 being cardinal set
for b3 being Relation-like Function-like Cardinal-yielding set holds
      b3 = b2 -powerfunc_of b1
   iff
      (for b4 being set holds
             b4 in proj1 b3
          iff
             b4 in b1 & b4 is cardinal set) &
       (for b4 being cardinal set
             st b4 in b1
          holds b3 . b4 = exp(b4,b2));

:: CARD_5:funcreg 2
registration
  let a1 be ordinal set;
  cluster alef a1 -> infinite;
end;

:: CARD_5:th 24
theorem
for b1 being infinite cardinal set holds
   ex b2 being ordinal set st
      b1 = alef b2;

:: CARD_5:th 25
theorem
for b1 being Element of NAT
for b2 being infinite cardinal set holds
   b2 <> 0 & b2 <> 1 & b2 <> 2 & b2 <> Card b1 & Card b1 in b2 & alef 0 c= b2;

:: CARD_5:th 26
theorem
for b1 being cardinal set
for b2 being infinite cardinal set
      st (b2 c= b1 or b2 in b1)
   holds b1 is infinite cardinal set;

:: CARD_5:th 27
theorem
for b1 being cardinal set
for b2 being infinite cardinal set
      st (b2 c= b1 or b2 in b1)
   holds b2 +` b1 = b1 & b1 +` b2 = b1 & b2 *` b1 = b1 & b1 *` b2 = b1;

:: CARD_5:th 28
theorem
for b1 being infinite cardinal set holds
   b1 +` b1 = b1 & b1 *` b1 = b1;

:: CARD_5:th 31
theorem
for b1 being cardinal set
for b2 being infinite cardinal set holds
   b1 c= exp(b1,b2);

:: CARD_5:th 32
theorem
for b1 being infinite cardinal set holds
   union b1 = b1;

:: CARD_5:funcreg 3
registration
  let a1 be infinite cardinal set;
  let a2 be cardinal set;
  cluster a1 +` a2 -> infinite cardinal;
end;

:: CARD_5:funcreg 4
registration
  let a1 be cardinal set;
  let a2 be infinite cardinal set;
  cluster a1 +` a2 -> infinite cardinal;
end;

:: CARD_5:funcreg 5
registration
  let a1, a2 be infinite cardinal set;
  cluster a1 *` a2 -> infinite cardinal;
end;

:: CARD_5:funcreg 6
registration
  let a1, a2 be infinite cardinal set;
  cluster exp(a1,a2) -> infinite cardinal;
end;

:: CARD_5:attrnot 1 => CARD_5:attr 1
definition
  let a1 be infinite cardinal set;
  attr a1 is regular means
    cf a1 = a1;
end;

:: CARD_5:dfs 3
definiens
  let a1 be infinite cardinal set;
To prove
     a1 is regular
it is sufficient to prove
  thus cf a1 = a1;

:: CARD_5:def 4
theorem
for b1 being infinite cardinal set holds
      b1 is regular
   iff
      cf b1 = b1;

:: CARD_5:attrnot 2 => CARD_5:attr 1
notation
  let a1 be infinite cardinal set;
  antonym irregular for regular;
end;

:: CARD_5:funcreg 7
registration
  let a1 be infinite cardinal set;
  cluster nextcard a1 -> infinite cardinal;
end;

:: CARD_5:th 34
theorem
cf alef 0 = alef 0;

:: CARD_5:th 35
theorem
for b1 being infinite cardinal set holds
   cf nextcard b1 = nextcard b1;

:: CARD_5:th 36
theorem
for b1 being infinite cardinal set holds
   alef 0 c= cf b1;

:: CARD_5:th 37
theorem
for b1 being Element of NAT holds
   cf 0 = 0 & cf Card (b1 + 1) = 1;

:: CARD_5:th 38
theorem
for b1 being set
for b2 being cardinal set
      st b1 c= b2 & Card b1 in cf b2
   holds sup b1 in b2 & union b1 in b2;

:: CARD_5:th 39
theorem
for b1, b2 being cardinal set
for b3 being Relation-like Function-like T-Sequence-like Ordinal-yielding set
      st proj1 b3 = b1 & proj2 b3 c= b2 & b1 in cf b2
   holds sup b3 in b2 & Union b3 in b2;

:: CARD_5:funcreg 8
registration
  let a1 be infinite cardinal set;
  cluster cf a1 -> infinite cardinal;
end;

:: CARD_5:th 40
theorem
for b1 being infinite cardinal set
      st cf b1 in b1
   holds b1 is limit;

:: CARD_5:th 41
theorem
for b1 being infinite cardinal set
      st cf b1 in b1
   holds ex b2 being Relation-like Function-like T-Sequence-like Ordinal-yielding set st
      proj1 b2 = cf b1 & proj2 b2 c= b1 & b2 is increasing & b1 = sup b2 & b2 is Relation-like Function-like Cardinal-yielding set & not 0 in proj2 b2;

:: CARD_5:th 42
theorem
for b1 being infinite cardinal set holds
   alef 0 is regular & nextcard b1 is regular;

:: CARD_5:th 43
theorem
for b1, b2 being infinite cardinal set
      st b1 c= b2
   holds exp(b1,b2) = exp(2,b2);

:: CARD_5:th 44
theorem
for b1, b2 being infinite cardinal set holds
exp(nextcard b1,b2) = (exp(b1,b2)) *` nextcard b1;

:: CARD_5:th 45
theorem
for b1, b2 being infinite cardinal set holds
Sum (b1 -powerfunc_of b2) c= exp(b2,b1);

:: CARD_5:th 46
theorem
for b1, b2 being infinite cardinal set
      st b1 is limit & b2 in cf b1
   holds exp(b1,b2) = Sum (b2 -powerfunc_of b1);

:: CARD_5:th 47
theorem
for b1, b2 being infinite cardinal set
      st cf b1 c= b2 & b2 in b1
   holds exp(b1,b2) = exp(Sum (b2 -powerfunc_of b1),cf b1);