
theorem Th1:
  for X being set, c,d being object
   st (ex a,b being object st a <> b & X = {a,b}) &
  c in X & d in X & c <> d holds X = {c,d}
proof
  let X be set, c,d be object such that
A1: ex a,b being object st a <> b & X = {a,b} and
A2: c in X and
A3: d in X and
A4: c <> d;
  consider a,b being object such that
  a <> b and
A5: X = {a,b} by A1;
A6: X c= {c,d}
  proof
A7: d = a or d = b by A3,A5,TARSKI:def 2;
A8: c = a or c = b by A2,A5,TARSKI:def 2;
    let x be object such that
A9: x in X;
    per cases by A5,A9,TARSKI:def 2;
    suppose
      x = a;
      hence thesis by A4,A8,A7,TARSKI:def 2;
    end;
    suppose
      x = b;
      hence thesis by A4,A8,A7,TARSKI:def 2;
   end;
  end;
  {c,d} c= X by A2,A3,ZFMISC_1:32;
  hence thesis by A6;
end;
