
theorem Th8:
  for A, e be set holds (e in TWOELEMENTSETS(A) iff
    (e is finite Subset of A &
  ex x,y being object st x in A & y in A & x<>y & e = {x,y} ))
proof
  let A,e be set;
  hereby
    assume e in TWOELEMENTSETS(A);
    then
A1: ex z being Subset of A st e=z & card z = 2;
    then reconsider e9=e as finite Subset of A;
    thus e is finite Subset of A by A1;
    consider x,y being object such that
A2: x<>y and
A3: e9={x,y} by A1,CARD_2:60;
    take x,y;
    x in e9 & y in e9 by A3,TARSKI:def 2;
    hence x in A & y in A;
    thus x<>y & e={x,y} by A2,A3;
  end;
  assume that
  e is finite Subset of A and
A4: ex x,y being object st x in A & y in A & x<>y & e={x,y};
  consider x,y being Element of A such that
A5: x in A and
  y in A and
A6: not x=y and
A7: e={x,y} by A4;
  reconsider xy = {x,y} as finite Subset of A by A5,ZFMISC_1:32;
  ex z being finite Subset of A st e=z & (card z)=2
  proof
    take xy;
    thus e=xy by A7;
    thus thesis by A6,CARD_2:57;
  end;
  hence thesis;
end;
