Article TARSKI, MML version 4.99.1005

:: TARSKI:th 2
theorem
for b1, b2 being set
      st for b3 being set holds
              b3 in b1
           iff
              b3 in b2
   holds b1 = b2;

:: TARSKI:funcnot 1 => TARSKI:func 1
definition
  let a1 be set;
  func {A1} -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 = a1;
end;

:: TARSKI:def 1
theorem
for b1, b2 being set holds
   b2 = {b1}
iff
   for b3 being set holds
         b3 in b2
      iff
         b3 = b1;

:: TARSKI:funcnot 2 => TARSKI:func 2
definition
  let a1, a2 be set;
  func {A1,A2} -> set means
    for b1 being set holds
          b1 in it
       iff
          (b1 = a1 or b1 = a2);
  commutativity;
::  for a1, a2 being set holds
::  {a1,a2} = {a2,a1};
end;

:: TARSKI:def 2
theorem
for b1, b2, b3 being set holds
   b3 = {b1,b2}
iff
   for b4 being set holds
         b4 in b3
      iff
         (b4 = b1 or b4 = b2);

:: TARSKI:prednot 1 => TARSKI:pred 1
definition
  let a1, a2 be set;
  pred A1 c= A2 means
    for b1 being set
          st b1 in a1
       holds b1 in a2;
  reflexivity;
::  for a1 being set holds
::     a1 c= a1;
end;

:: TARSKI:dfs 3
definiens
  let a1, a2 be set;
To prove
     a1 c= a2
it is sufficient to prove
  thus for b1 being set
          st b1 in a1
       holds b1 in a2;

:: TARSKI:def 3
theorem
for b1, b2 being set holds
   b1 c= b2
iff
   for b3 being set
         st b3 in b1
      holds b3 in b2;

:: TARSKI:funcnot 3 => TARSKI:func 3
definition
  let a1 be set;
  func union A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          ex b2 being set st
             b1 in b2 & b2 in a1;
end;

:: TARSKI:def 4
theorem
for b1, b2 being set holds
   b2 = union b1
iff
   for b3 being set holds
         b3 in b2
      iff
         ex b4 being set st
            b3 in b4 & b4 in b1;

:: TARSKI:th 7
theorem
for b1, b2 being set
      st b1 in b2
   holds ex b3 being set st
      b3 in b2 &
       (for b4 being set
             st b4 in b2
          holds not b4 in b3);

:: TARSKI:sch 1
scheme TARSKI:sch 1
{F1 -> set}:
ex b1 being set st
   for b2 being set holds
         b2 in b1
      iff
         ex b3 being set st
            b3 in F1() & P1[b3, b2]
provided
   for b1, b2, b3 being set
         st P1[b1, b2] & P1[b1, b3]
      holds b2 = b3;


:: TARSKI:funcnot 4 => TARSKI:func 4
definition
  let a1, a2 be set;
  func [A1,A2] -> set equals
    {{a1,a2},{a1}};
end;

:: TARSKI:def 5
theorem
for b1, b2 being set holds
[b1,b2] = {{b1,b2},{b1}};

:: TARSKI:prednot 2 => TARSKI:pred 2
definition
  let a1, a2 be set;
  pred A1,A2 are_equipotent means
    ex b1 being set st
       (for b2 being set
              st b2 in a1
           holds ex b3 being set st
              b3 in a2 & [b2,b3] in b1) &
        (for b2 being set
              st b2 in a2
           holds ex b3 being set st
              b3 in a1 & [b3,b2] in b1) &
        (for b2, b3, b4, b5 being set
              st [b2,b3] in b1 & [b4,b5] in b1
           holds    b2 = b4
           iff
              b3 = b5);
end;

:: TARSKI:dfs 6
definiens
  let a1, a2 be set;
To prove
     a1,a2 are_equipotent
it is sufficient to prove
  thus ex b1 being set st
       (for b2 being set
              st b2 in a1
           holds ex b3 being set st
              b3 in a2 & [b2,b3] in b1) &
        (for b2 being set
              st b2 in a2
           holds ex b3 being set st
              b3 in a1 & [b3,b2] in b1) &
        (for b2, b3, b4, b5 being set
              st [b2,b3] in b1 & [b4,b5] in b1
           holds    b2 = b4
           iff
              b3 = b5);

:: TARSKI:def 6
theorem
for b1, b2 being set holds
   b1,b2 are_equipotent
iff
   ex b3 being set st
      (for b4 being set
             st b4 in b1
          holds ex b5 being set st
             b5 in b2 & [b4,b5] in b3) &
       (for b4 being set
             st b4 in b2
          holds ex b5 being set st
             b5 in b1 & [b5,b4] in b3) &
       (for b4, b5, b6, b7 being set
             st [b4,b5] in b3 & [b6,b7] in b3
          holds    b4 = b6
          iff
             b5 = b7);

:: TARSKI:th 9
theorem
for b1 being set holds
   ex b2 being set st
      b1 in b2 &
       (for b3, b4 being set
             st b3 in b2 & b4 c= b3
          holds b4 in b2) &
       (for b3 being set
             st b3 in b2
          holds ex b4 being set st
             b4 in b2 &
              (for b5 being set
                    st b5 c= b3
                 holds b5 in b4)) &
       (for b3 being set
             st b3 c= b2 & not b3,b2 are_equipotent
          holds b3 in b2);