
theorem Lemma5:
  for a, b being object st a <> b holds
    {[a, b]} is asymmetric
  proof
    let a, b be object;
    assume
A0: a <> b;
    set R = {[a, b]};
    for x, y being object st [x, y] in R holds not [y, x] in R
    proof
      let x, y be object;
      assume [x, y] in R; then
      [x, y] = [a, b] by TARSKI:def 1; then
A1:   x = a & y = b by XTUPLE_0:1;
      assume [y, x] in R; then
      [y, x] = [a, b] by TARSKI:def 1;
      hence contradiction by A0, A1,XTUPLE_0:1;
    end;
    hence thesis by LemAsym;
  end;
