
theorem Lemma8:
  for a, b being set holds
    a <> b iff {[a, b]} misses {[a, a], [b, b]}
  proof
    let a, b be set;
    thus a <> b implies {[a, b]} misses {[a, a], [b, b]}
    proof
      assume
A0:   a <> b; then
A1:   [a,b] <> [a,a] by XTUPLE_0:1;
A2:   [a,b] <> [b,b] by A0,XTUPLE_0:1;
      set A = {[a, b]}, B = {[a, a], [b, b]};
      assume A meets B; then
      consider x being object such that
A4:   x in A & x in B by XBOOLE_0:3;
      x = [a, b] by TARSKI:def 1,A4;
      hence thesis by A1, A2,A4, TARSKI:def 2;
    end;
    assume
A0: {[a, b]} misses {[a, a], [b, b]};
    assume a = b; then
    not [a,a] in {[a, a], [b, b]} by ZFMISC_1:48,A0;
    hence thesis by TARSKI:def 2;
  end;
