reserve X,x,y,z for set;
reserve n,m,k,k9,d9 for Nat;
reserve d for non zero Nat;
reserve i,i0,i1 for Element of Seg d;

theorem Th5:
  card X = 2 iff
  ex x,y st x in X & y in X & x <> y & for z st z in X holds z = x or z = y
proof
  hereby
    assume
A1: card X = 2;
    then reconsider X9 = X as finite set;
    consider x,y being object such that
A2: x <> y and
A3: X9 = {x,y} by A1,CARD_2:60;
     reconsider x,y as set by TARSKI:1;
    take x,y;
    thus x in X & y in X & x <> y &
    for z st z in X holds z = x or z = y by A2,A3,TARSKI:def 2;
  end;
  given x,y such that
A4: x in X and
A5: y in X and
A6: x <> y and
A7: for z st z in X holds z = x or z = y;
  for z be object holds z in X iff z = x or z = y by A4,A5,A7;
  then X = {x,y} by TARSKI:def 2;
  hence thesis by A6,CARD_2:57;
end;
