
theorem Th8:
  for I being 2-element set, i,j being Element of I st i <> j holds I = {i,j}
proof
  let I be 2-element set;
  let i,j be Element of I;
  assume A1: i <> j;
  for x being object holds x = i or x = j iff x in I
  proof
    let x be object;
    thus x = i or x = j implies x in I;
    assume A2: x in I;
    assume x <> i & x <> j;
    then A3: card {x,i,j} = 3 by A1, CARD_2:58;
    {x,i,j} c= I
    proof
      let z be object;
      assume z in {x,i,j};
      then z = x or z = i or z = j by ENUMSET1:def 1;
      hence thesis by A2;
    end;
    then card {x,i,j} c= card I by CARD_1:11;
    then A4: {0,1,2} c= 2 by A3, CARD_1:def 7, CARD_1:51;
    2 in {0,1,2} by ENUMSET1:def 1;
    then 2 in 2 by A4;
    hence contradiction;
  end;
  hence I = {i,j} by TARSKI:def 2;
end;
