Article CARD_FIL, MML version 4.99.1005

:: CARD_FIL:th 1
theorem
for b1 being set
for b2 being infinite set holds
   Card {b1} in Card b2;

:: CARD_FIL:funcreg 1
registration
  let a1 be infinite set;
  cluster Card a1 -> infinite cardinal;
end;

:: CARD_FIL:sch 1
scheme CARD_FIL:sch 1
{F1 -> non empty set,
  F2 -> set}:
P1[F2()]
provided
   F2() in {b1 where b1 is Element of F1(): P1[b1]};


:: CARD_FIL:th 2
theorem
for b1 being non empty set holds
   {b1} is non empty Element of bool bool b1 &
    not {} in {b1} &
    (for b2, b3 being Element of bool b1 holds
    (b2 in {b1} & b3 in {b1} implies b2 /\ b3 in {b1}) &
     (b2 in {b1} & b2 c= b3 implies b3 in {b1}));

:: CARD_FIL:modenot 1 => CARD_FIL:mode 1
definition
  let a1 be non empty set;
  mode Filter of A1 -> non empty Element of bool bool a1 means
    not {} in it &
     (for b1, b2 being Element of bool a1 holds
     (b1 in it & b2 in it implies b1 /\ b2 in it) & (b1 in it & b1 c= b2 implies b2 in it));
end;

:: CARD_FIL:dfs 1
definiens
  let a1 be non empty set;
  let a2 be non empty Element of bool bool a1;
To prove
     a2 is Filter of a1
it is sufficient to prove
  thus not {} in a2 &
     (for b1, b2 being Element of bool a1 holds
     (b1 in a2 & b2 in a2 implies b1 /\ b2 in a2) &
      (b1 in a2 & b1 c= b2 implies b2 in a2));

:: CARD_FIL:def 1
theorem
for b1 being non empty set
for b2 being non empty Element of bool bool b1 holds
      b2 is Filter of b1
   iff
      not {} in b2 &
       (for b3, b4 being Element of bool b1 holds
       (b3 in b2 & b4 in b2 implies b3 /\ b4 in b2) &
        (b3 in b2 & b3 c= b4 implies b4 in b2));

:: CARD_FIL:th 3
theorem
for b1 being non empty set
for b2 being set holds
      b2 is Filter of b1
   iff
      b2 is non empty Element of bool bool b1 &
       not {} in b2 &
       (for b3, b4 being Element of bool b1 holds
       (b3 in b2 & b4 in b2 implies b3 /\ b4 in b2) &
        (b3 in b2 & b3 c= b4 implies b4 in b2));

:: CARD_FIL:th 4
theorem
for b1 being non empty set holds
   {b1} is Filter of b1;

:: CARD_FIL:th 5
theorem
for b1 being non empty set
for b2 being Filter of b1 holds
   b1 in b2;

:: CARD_FIL:th 6
theorem
for b1 being non empty set
for b2 being Element of bool b1
for b3 being Filter of b1
      st b2 in b3
   holds not b1 \ b2 in b3;

:: CARD_FIL:th 7
theorem
for b1 being non empty set
for b2 being Filter of b1
for b3 being non empty Element of bool bool b1
      st for b4 being Element of bool b1 holds
              b4 in b3
           iff
              b4 ` in b2
   holds not b1 in b3 &
    (for b4, b5 being Element of bool b1 holds
    (b4 in b3 & b5 in b3 implies b4 \/ b5 in b3) &
     (b4 in b3 & b5 c= b4 implies b5 in b3));

:: CARD_FIL:funcnot 1 => SETFAM_1:func 7
notation
  let a1 be non empty set;
  let a2 be Element of bool bool a1;
  synonym dual a2 for COMPLEMENT a2;
end;

:: CARD_FIL:funcreg 2
registration
  let a1 be non empty set;
  let a2 be non empty Element of bool bool a1;
  cluster COMPLEMENT a2 -> non empty;
end;

:: CARD_FIL:th 8
theorem
for b1 being non empty set
for b2 being non empty Element of bool bool b1 holds
   COMPLEMENT b2 = {b3 where b3 is Element of bool b1: b3 ` in b2};

:: CARD_FIL:th 9
theorem
for b1 being non empty set
for b2 being non empty Element of bool bool b1 holds
   COMPLEMENT b2 = {b3 ` where b3 is Element of bool b1: b3 in b2};

:: CARD_FIL:modenot 2 => CARD_FIL:mode 2
definition
  let a1 be non empty set;
  mode Ideal of A1 -> non empty Element of bool bool a1 means
    not a1 in it &
     (for b1, b2 being Element of bool a1 holds
     (b1 in it & b2 in it implies b1 \/ b2 in it) & (b1 in it & b2 c= b1 implies b2 in it));
end;

:: CARD_FIL:dfs 2
definiens
  let a1 be non empty set;
  let a2 be non empty Element of bool bool a1;
To prove
     a2 is Ideal of a1
it is sufficient to prove
  thus not a1 in a2 &
     (for b1, b2 being Element of bool a1 holds
     (b1 in a2 & b2 in a2 implies b1 \/ b2 in a2) &
      (b1 in a2 & b2 c= b1 implies b2 in a2));

:: CARD_FIL:def 2
theorem
for b1 being non empty set
for b2 being non empty Element of bool bool b1 holds
      b2 is Ideal of b1
   iff
      not b1 in b2 &
       (for b3, b4 being Element of bool b1 holds
       (b3 in b2 & b4 in b2 implies b3 \/ b4 in b2) &
        (b3 in b2 & b4 c= b3 implies b4 in b2));

:: CARD_FIL:funcnot 2 => CARD_FIL:func 1
definition
  let a1 be non empty set;
  let a2 be Filter of a1;
  redefine func dual a2 -> Ideal of a1;
  involutiveness;
::  for a1 being non empty set
::  for a2 being Filter of a1 holds
::     dual dual a2 = a2;
end;

:: CARD_FIL:th 10
theorem
for b1 being non empty set
for b2 being Filter of b1
for b3 being Ideal of b1 holds
   (for b4 being Element of bool b1
          st b4 in b2
       holds not b4 in dual b2) &
    (for b4 being Element of bool b1
          st b4 in b3
       holds not b4 in COMPLEMENT b3);

:: CARD_FIL:th 11
theorem
for b1 being non empty set
for b2 being Ideal of b1 holds
   {} in b2;

:: CARD_FIL:prednot 1 => CARD_FIL:pred 1
definition
  let a1 be non empty set;
  let a2 be cardinal set;
  let a3 be non empty Element of bool bool a1;
  pred A3 is_multiplicative_with A2 means
    for b1 being non empty set
          st b1 c= a3 & Card b1 in a2
       holds meet b1 in a3;
end;

:: CARD_FIL:dfs 3
definiens
  let a1 be non empty set;
  let a2 be cardinal set;
  let a3 be non empty Element of bool bool a1;
To prove
     a3 is_multiplicative_with a2
it is sufficient to prove
  thus for b1 being non empty set
          st b1 c= a3 & Card b1 in a2
       holds meet b1 in a3;

:: CARD_FIL:def 3
theorem
for b1 being non empty set
for b2 being cardinal set
for b3 being non empty Element of bool bool b1 holds
      b3 is_multiplicative_with b2
   iff
      for b4 being non empty set
            st b4 c= b3 & Card b4 in b2
         holds meet b4 in b3;

:: CARD_FIL:prednot 2 => CARD_FIL:pred 2
definition
  let a1 be non empty set;
  let a2 be cardinal set;
  let a3 be non empty Element of bool bool a1;
  pred A3 is_additive_with A2 means
    for b1 being non empty set
          st b1 c= a3 & Card b1 in a2
       holds union b1 in a3;
end;

:: CARD_FIL:dfs 4
definiens
  let a1 be non empty set;
  let a2 be cardinal set;
  let a3 be non empty Element of bool bool a1;
To prove
     a3 is_additive_with a2
it is sufficient to prove
  thus for b1 being non empty set
          st b1 c= a3 & Card b1 in a2
       holds union b1 in a3;

:: CARD_FIL:def 4
theorem
for b1 being non empty set
for b2 being cardinal set
for b3 being non empty Element of bool bool b1 holds
      b3 is_additive_with b2
   iff
      for b4 being non empty set
            st b4 c= b3 & Card b4 in b2
         holds union b4 in b3;

:: CARD_FIL:prednot 3 => CARD_FIL:pred 1
notation
  let a1 be non empty set;
  let a2 be cardinal set;
  let a3 be Filter of a1;
  synonym a3 is_complete_with a2 for a3 is_multiplicative_with a2;
end;

:: CARD_FIL:prednot 4 => CARD_FIL:pred 2
notation
  let a1 be non empty set;
  let a2 be cardinal set;
  let a3 be Ideal of a1;
  synonym a3 is_complete_with a2 for a3 is_additive_with a2;
end;

:: CARD_FIL:th 12
theorem
for b1 being cardinal set
for b2 being non empty set
for b3 being non empty Element of bool bool b2
      st b3 is_multiplicative_with b1
   holds COMPLEMENT b3 is_additive_with b1;

:: CARD_FIL:attrnot 1 => CARD_FIL:attr 1
definition
  let a1 be non empty set;
  let a2 be Filter of a1;
  attr a2 is uniform means
    for b1 being Element of bool a1
          st b1 in a2
       holds Card b1 = Card a1;
end;

:: CARD_FIL:dfs 5
definiens
  let a1 be non empty set;
  let a2 be Filter of a1;
To prove
     a2 is uniform
it is sufficient to prove
  thus for b1 being Element of bool a1
          st b1 in a2
       holds Card b1 = Card a1;

:: CARD_FIL:def 5
theorem
for b1 being non empty set
for b2 being Filter of b1 holds
      b2 is uniform(b1)
   iff
      for b3 being Element of bool b1
            st b3 in b2
         holds Card b3 = Card b1;

:: CARD_FIL:attrnot 2 => CARD_FIL:attr 2
definition
  let a1 be non empty set;
  let a2 be Filter of a1;
  attr a2 is principal means
    ex b1 being Element of bool a1 st
       b1 in a2 &
        (for b2 being Element of bool a1
              st b2 in a2
           holds b1 c= b2);
end;

:: CARD_FIL:dfs 6
definiens
  let a1 be non empty set;
  let a2 be Filter of a1;
To prove
     a2 is principal
it is sufficient to prove
  thus ex b1 being Element of bool a1 st
       b1 in a2 &
        (for b2 being Element of bool a1
              st b2 in a2
           holds b1 c= b2);

:: CARD_FIL:def 6
theorem
for b1 being non empty set
for b2 being Filter of b1 holds
      b2 is principal(b1)
   iff
      ex b3 being Element of bool b1 st
         b3 in b2 &
          (for b4 being Element of bool b1
                st b4 in b2
             holds b3 c= b4);

:: CARD_FIL:attrnot 3 => CARD_FIL:attr 3
definition
  let a1 be non empty set;
  let a2 be Filter of a1;
  attr a2 is being_ultrafilter means
    for b1 being Element of bool a1
          st not b1 in a2
       holds a1 \ b1 in a2;
end;

:: CARD_FIL:dfs 7
definiens
  let a1 be non empty set;
  let a2 be Filter of a1;
To prove
     a2 is being_ultrafilter
it is sufficient to prove
  thus for b1 being Element of bool a1
          st not b1 in a2
       holds a1 \ b1 in a2;

:: CARD_FIL:def 7
theorem
for b1 being non empty set
for b2 being Filter of b1 holds
      b2 is being_ultrafilter(b1)
   iff
      for b3 being Element of bool b1
            st not b3 in b2
         holds b1 \ b3 in b2;

:: CARD_FIL:funcnot 3 => CARD_FIL:func 2
definition
  let a1 be non empty set;
  let a2 be Filter of a1;
  let a3 be Element of bool a1;
  func Extend_Filter(A2,A3) -> non empty Element of bool bool a1 equals
    {b1 where b1 is Element of bool a1: ex b2 being Element of bool a1 st
       b2 in {b3 /\ a3 where b3 is Element of bool a1: b3 in a2} &
        b2 c= b1};
end;

:: CARD_FIL:def 8
theorem
for b1 being non empty set
for b2 being Filter of b1
for b3 being Element of bool b1 holds
   Extend_Filter(b2,b3) = {b4 where b4 is Element of bool b1: ex b5 being Element of bool b1 st
      b5 in {b6 /\ b3 where b6 is Element of bool b1: b6 in b2} &
       b5 c= b4};

:: CARD_FIL:th 13
theorem
for b1 being non empty set
for b2 being Element of bool b1
for b3 being Filter of b1
for b4 being Element of bool b1 holds
      b4 in Extend_Filter(b3,b2)
   iff
      ex b5 being Element of bool b1 st
         b5 in b3 & b5 /\ b2 c= b4;

:: CARD_FIL:th 14
theorem
for b1 being non empty set
for b2 being Element of bool b1
for b3 being Filter of b1
      st for b4 being Element of bool b1
              st b4 in b3
           holds b4 meets b2
   holds b2 in Extend_Filter(b3,b2) & Extend_Filter(b3,b2) is Filter of b1 & b3 c= Extend_Filter(b3,b2);

:: CARD_FIL:funcnot 4 => CARD_FIL:func 3
definition
  let a1 be non empty set;
  func Filters A1 -> non empty Element of bool bool bool a1 equals
    {b1 where b1 is Element of bool bool a1: b1 is Filter of a1};
end;

:: CARD_FIL:def 9
theorem
for b1 being non empty set holds
   Filters b1 = {b2 where b2 is Element of bool bool b1: b2 is Filter of b1};

:: CARD_FIL:th 15
theorem
for b1 being non empty set
for b2 being set holds
      b2 in Filters b1
   iff
      b2 is Filter of b1;

:: CARD_FIL:th 16
theorem
for b1 being non empty set
for b2 being non empty Element of bool Filters b1
      st b2 is c=-linear
   holds union b2 is Filter of b1;

:: CARD_FIL:th 17
theorem
for b1 being non empty set
for b2 being Filter of b1 holds
   ex b3 being Filter of b1 st
      b2 c= b3 & b3 is being_ultrafilter(b1);

:: CARD_FIL:funcnot 5 => CARD_FIL:func 4
definition
  let a1 be infinite set;
  func Frechet_Filter A1 -> Filter of a1 equals
    {b1 where b1 is Element of bool a1: Card (a1 \ b1) in Card a1};
end;

:: CARD_FIL:def 10
theorem
for b1 being infinite set holds
   Frechet_Filter b1 = {b2 where b2 is Element of bool b1: Card (b1 \ b2) in Card b1};

:: CARD_FIL:funcnot 6 => CARD_FIL:func 5
definition
  let a1 be infinite set;
  func Frechet_Ideal A1 -> Ideal of a1 equals
    dual Frechet_Filter a1;
end;

:: CARD_FIL:def 11
theorem
for b1 being infinite set holds
   Frechet_Ideal b1 = dual Frechet_Filter b1;

:: CARD_FIL:th 18
theorem
for b1 being infinite set
for b2 being Element of bool b1 holds
      b2 in Frechet_Filter b1
   iff
      Card (b1 \ b2) in Card b1;

:: CARD_FIL:th 19
theorem
for b1 being infinite set
for b2 being Element of bool b1 holds
      b2 in Frechet_Ideal b1
   iff
      Card b2 in Card b1;

:: CARD_FIL:th 20
theorem
for b1 being infinite set
for b2 being Filter of b1
      st Frechet_Filter b1 c= b2
   holds b2 is uniform(b1);

:: CARD_FIL:th 21
theorem
for b1 being infinite set
for b2 being Filter of b1
      st b2 is uniform(b1) & b2 is being_ultrafilter(b1)
   holds Frechet_Filter b1 c= b2;

:: CARD_FIL:exreg 1
registration
  let a1 be infinite set;
  cluster non empty non principal being_ultrafilter Filter of a1;
end;

:: CARD_FIL:condreg 1
registration
  let a1 be infinite set;
  cluster uniform being_ultrafilter -> non principal (Filter of a1);
end;

:: CARD_FIL:th 22
theorem
for b1 being infinite set
for b2 being being_ultrafilter Filter of b1
for b3 being Element of bool b1 holds
      b3 in b2
   iff
      not b3 in dual b2;

:: CARD_FIL:th 23
theorem
for b1 being infinite set
for b2 being Filter of b1
      st b2 is not principal(b1) & b2 is being_ultrafilter(b1) & b2 is_multiplicative_with Card b1
   holds b2 is uniform(b1);

:: CARD_FIL:th 24
theorem
for b1 being cardinal set holds
   nextcard b1 c= exp(2,b1);

:: CARD_FIL:prednot 5 => CARD_FIL:pred 3
definition
  pred GCH means
    for b1 being infinite cardinal set holds
       nextcard b1 = exp(2,b1);
end;

:: CARD_FIL:dfs 12
definiens
To prove
     GCH
it is sufficient to prove
  thus for b1 being infinite cardinal set holds
       nextcard b1 = exp(2,b1);

:: CARD_FIL:def 12
theorem
   (GCH
iff
   for b1 being infinite cardinal set holds
      nextcard b1 = exp(2,b1));

:: CARD_FIL:attrnot 4 => CARD_FIL:attr 4
definition
  let a1 be infinite cardinal set;
  attr a1 is inaccessible means
    a1 is regular & a1 is limit;
end;

:: CARD_FIL:dfs 13
definiens
  let a1 be infinite cardinal set;
To prove
     a1 is inaccessible
it is sufficient to prove
  thus a1 is regular & a1 is limit;

:: CARD_FIL:def 13
theorem
for b1 being infinite cardinal set holds
      b1 is inaccessible
   iff
      b1 is regular & b1 is limit;

:: CARD_FIL:prednot 6 => CARD_FIL:attr 4
notation
  let a1 be infinite cardinal set;
  synonym a1 is_inaccessible_cardinal for inaccessible;
end;

:: CARD_FIL:condreg 2
registration
  cluster infinite cardinal inaccessible -> limit regular (set);
end;

:: CARD_FIL:th 25
theorem
alef {} is inaccessible;

:: CARD_FIL:attrnot 5 => CARD_FIL:attr 5
definition
  let a1 be infinite cardinal set;
  attr a1 is strong_limit means
    for b1 being cardinal set
          st b1 in a1
       holds exp(2,b1) in a1;
end;

:: CARD_FIL:dfs 14
definiens
  let a1 be infinite cardinal set;
To prove
     a1 is strong_limit
it is sufficient to prove
  thus for b1 being cardinal set
          st b1 in a1
       holds exp(2,b1) in a1;

:: CARD_FIL:def 14
theorem
for b1 being infinite cardinal set holds
      b1 is strong_limit
   iff
      for b2 being cardinal set
            st b2 in b1
         holds exp(2,b2) in b1;

:: CARD_FIL:prednot 7 => CARD_FIL:attr 5
notation
  let a1 be infinite cardinal set;
  synonym a1 is_strong_limit_cardinal for strong_limit;
end;

:: CARD_FIL:th 26
theorem
alef {} is strong_limit;

:: CARD_FIL:th 27
theorem
for b1 being infinite cardinal set
      st b1 is strong_limit
   holds b1 is limit;

:: CARD_FIL:condreg 3
registration
  cluster infinite cardinal strong_limit -> limit (set);
end;

:: CARD_FIL:th 28
theorem
for b1 being infinite cardinal set
      st GCH & b1 is limit
   holds b1 is strong_limit;

:: CARD_FIL:attrnot 6 => CARD_FIL:attr 6
definition
  let a1 be infinite cardinal set;
  attr a1 is strongly_inaccessible means
    a1 is regular & a1 is strong_limit;
end;

:: CARD_FIL:dfs 15
definiens
  let a1 be infinite cardinal set;
To prove
     a1 is strongly_inaccessible
it is sufficient to prove
  thus a1 is regular & a1 is strong_limit;

:: CARD_FIL:def 15
theorem
for b1 being infinite cardinal set holds
      b1 is strongly_inaccessible
   iff
      b1 is regular & b1 is strong_limit;

:: CARD_FIL:prednot 8 => CARD_FIL:attr 6
notation
  let a1 be infinite cardinal set;
  synonym a1 is_strongly_inaccessible_cardinal for strongly_inaccessible;
end;

:: CARD_FIL:condreg 4
registration
  cluster infinite cardinal strongly_inaccessible -> regular strong_limit (set);
end;

:: CARD_FIL:th 29
theorem
alef {} is strongly_inaccessible;

:: CARD_FIL:th 30
theorem
for b1 being infinite cardinal set
      st b1 is strongly_inaccessible
   holds b1 is inaccessible;

:: CARD_FIL:condreg 5
registration
  cluster infinite cardinal strongly_inaccessible -> inaccessible (set);
end;

:: CARD_FIL:th 31
theorem
for b1 being infinite cardinal set
      st GCH & b1 is inaccessible
   holds b1 is strongly_inaccessible;

:: CARD_FIL:attrnot 7 => CARD_FIL:attr 7
definition
  let a1 be infinite cardinal set;
  attr a1 is measurable means
    ex b1 being Filter of a1 st
       b1 is_multiplicative_with a1 & b1 is not principal(a1) & b1 is being_ultrafilter(a1);
end;

:: CARD_FIL:dfs 16
definiens
  let a1 be infinite cardinal set;
To prove
     a1 is measurable
it is sufficient to prove
  thus ex b1 being Filter of a1 st
       b1 is_multiplicative_with a1 & b1 is not principal(a1) & b1 is being_ultrafilter(a1);

:: CARD_FIL:def 16
theorem
for b1 being infinite cardinal set holds
      b1 is measurable
   iff
      ex b2 being Filter of b1 st
         b2 is_multiplicative_with b1 & b2 is not principal(b1) & b2 is being_ultrafilter(b1);

:: CARD_FIL:prednot 9 => CARD_FIL:attr 7
notation
  let a1 be infinite cardinal set;
  synonym a1 is_measurable_cardinal for measurable;
end;

:: CARD_FIL:th 32
theorem
for b1 being ordinal being_limit_ordinal set
for b2 being set
      st b2 c= b1 & sup b2 = b1
   holds union b2 = b1;

:: CARD_FIL:th 33
theorem
for b1 being infinite cardinal set
      st b1 is measurable
   holds b1 is regular;

:: CARD_FIL:funcreg 3
registration
  let a1 be infinite cardinal set;
  cluster nextcard a1 -> cardinal non limit;
end;

:: CARD_FIL:exreg 2
registration
  cluster epsilon-transitive epsilon-connected ordinal infinite cardinal non limit set;
end;

:: CARD_FIL:condreg 6
registration
  cluster infinite cardinal non limit -> regular (set);
end;

:: CARD_FIL:funcnot 7 => CARD_FIL:func 6
definition
  let a1 be cardinal non limit set;
  func predecessor A1 -> cardinal set means
    a1 = nextcard it;
end;

:: CARD_FIL:def 17
theorem
for b1 being cardinal non limit set
for b2 being cardinal set holds
      b2 = predecessor b1
   iff
      b1 = nextcard b2;

:: CARD_FIL:funcreg 4
registration
  let a1 be infinite cardinal non limit set;
  cluster predecessor a1 -> infinite cardinal;
end;

:: CARD_FIL:modenot 3
definition
  let a1 be set;
  let a2, a3 be cardinal set;
  mode Inf_Matrix of a2,a3,a1 is Function-like quasi_total Relation of [:a2,a3:],a1;
end;

:: CARD_FIL:prednot 10 => CARD_FIL:pred 4
definition
  let a1 be infinite cardinal non limit set;
  let a2 be Function-like quasi_total Relation of [:predecessor a1,a1:],bool a1;
  pred A2 is_Ulam_Matrix_of A1 means
    (for b1 being Element of predecessor a1
     for b2, b3 being Element of a1
           st b2 <> b3
        holds (a2 .(b1,b2)) /\ (a2 .(b1,b3)) is empty) &
     (for b1 being Element of a1
     for b2, b3 being Element of predecessor a1
           st b2 <> b3
        holds (a2 .(b2,b1)) /\ (a2 .(b3,b1)) is empty) &
     (for b1 being Element of predecessor a1 holds
        Card (a1 \ union {a2 .(b1,b2) where b2 is Element of a1: b2 in a1}) c= predecessor a1) &
     (for b1 being Element of a1 holds
        Card (a1 \ union {a2 .(b2,b1) where b2 is Element of predecessor a1: b2 in predecessor a1}) c= predecessor a1);
end;

:: CARD_FIL:dfs 18
definiens
  let a1 be infinite cardinal non limit set;
  let a2 be Function-like quasi_total Relation of [:predecessor a1,a1:],bool a1;
To prove
     a2 is_Ulam_Matrix_of a1
it is sufficient to prove
  thus (for b1 being Element of predecessor a1
     for b2, b3 being Element of a1
           st b2 <> b3
        holds (a2 .(b1,b2)) /\ (a2 .(b1,b3)) is empty) &
     (for b1 being Element of a1
     for b2, b3 being Element of predecessor a1
           st b2 <> b3
        holds (a2 .(b2,b1)) /\ (a2 .(b3,b1)) is empty) &
     (for b1 being Element of predecessor a1 holds
        Card (a1 \ union {a2 .(b1,b2) where b2 is Element of a1: b2 in a1}) c= predecessor a1) &
     (for b1 being Element of a1 holds
        Card (a1 \ union {a2 .(b2,b1) where b2 is Element of predecessor a1: b2 in predecessor a1}) c= predecessor a1);

:: CARD_FIL:def 18
theorem
for b1 being infinite cardinal non limit set
for b2 being Function-like quasi_total Relation of [:predecessor b1,b1:],bool b1 holds
      b2 is_Ulam_Matrix_of b1
   iff
      (for b3 being Element of predecessor b1
       for b4, b5 being Element of b1
             st b4 <> b5
          holds (b2 .(b3,b4)) /\ (b2 .(b3,b5)) is empty) &
       (for b3 being Element of b1
       for b4, b5 being Element of predecessor b1
             st b4 <> b5
          holds (b2 .(b4,b3)) /\ (b2 .(b5,b3)) is empty) &
       (for b3 being Element of predecessor b1 holds
          Card (b1 \ union {b2 .(b3,b4) where b4 is Element of b1: b4 in b1}) c= predecessor b1) &
       (for b3 being Element of b1 holds
          Card (b1 \ union {b2 .(b4,b3) where b4 is Element of predecessor b1: b4 in predecessor b1}) c= predecessor b1);

:: CARD_FIL:th 34
theorem
for b1 being infinite cardinal non limit set holds
   ex b2 being Function-like quasi_total Relation of [:predecessor b1,b1:],bool b1 st
      b2 is_Ulam_Matrix_of b1;

:: CARD_FIL:th 35
theorem
for b1 being infinite cardinal non limit set
for b2 being Ideal of b1
      st b2 is_additive_with b1 & Frechet_Ideal b1 c= b2
   holds ex b3 being Element of bool bool b1 st
      Card b3 = b1 &
       (for b4 being set
             st b4 in b3
          holds not b4 in b2) &
       (for b4, b5 being set
             st b4 in b3 & b5 in b3 & b4 <> b5
          holds b4 misses b5);

:: CARD_FIL:th 36
theorem
for b1 being set
for b2 being cardinal set
      st b2 c= Card b1
   holds ex b3 being set st
      b3 c= b1 & Card b3 = b2;

:: CARD_FIL:th 37
theorem
for b1 being infinite cardinal non limit set
for b2 being Filter of b1
      st b2 is uniform(b1) & b2 is being_ultrafilter(b1)
   holds not b2 is_multiplicative_with b1;

:: CARD_FIL:th 38
theorem
for b1 being infinite cardinal set
      st b1 is measurable
   holds b1 is limit;

:: CARD_FIL:th 39
theorem
for b1 being infinite cardinal set
      st b1 is measurable
   holds b1 is inaccessible;

:: CARD_FIL:th 40
theorem
for b1 being infinite cardinal set
      st b1 is measurable
   holds b1 is strong_limit;

:: CARD_FIL:th 41
theorem
for b1 being infinite cardinal set
      st b1 is measurable
   holds b1 is strongly_inaccessible;