theorem
  x <> y implies { x } \+\ { y } = { x,y }
proof
  assume
A1: x <> y;
  for z holds z in { x } \+\ { y } iff z=x or z=y
  proof
    let z;
A2: z in {y} iff z=y by TARSKI:def 1;
    z in {x} iff z=x by TARSKI:def 1;
    hence thesis by A1,A2,XBOOLE_0:1;
  end;
  hence thesis by TARSKI:def 2;
end;
