
theorem Lemma9:
  for a, b being object st a <> b holds
    {[a, b], [b, a]} is irreflexive symmetric
  proof
    let a, b be object;
    assume
A0: a <> b;
    reconsider R = {[a, b], [b, a]} as Relation;
a1: for x, y being object st [x,y] in R holds [y,x] in R
    proof
      let x, y be object;
      assume [x,y] in R; then
      [x,y] = [a,b] or [x,y] = [b,a] by TARSKI:def 2; then
      x = a & y = b or x = b & y = a by XTUPLE_0:1;
      hence thesis by TARSKI:def 2;
    end;
    for x being object st x in field R holds not [x, x] in R
    proof
      let x be object;
      assume x in field R & [x, x] in R; then
      [x, x] = [a, b] or [x, x] = [b, a] by TARSKI:def 2; then
      x = a & x = b or x = b & x = a by XTUPLE_0:1;
      hence contradiction by A0;
    end;
    hence thesis by a1,RELAT_2:def 10,RELAT_2:def 2,LemSym;
  end;
