
theorem Th1:
  for x,y,z being set st x <> z & y <> z holds {x,y} \ {z} = {x,y}
proof
  let x,y,z be set;
  assume that
A1: x <> z and
A2: y <> z;
  for a being object st a in {x,y} holds not a in {z}
  proof
    let a be object;
    assume a in {x,y};
    then a <> z by A1,A2,TARSKI:def 2;
    hence thesis by TARSKI:def 1;
  end;
  then {x,y} misses {z} by XBOOLE_0:3;
  hence thesis by XBOOLE_1:83;
end;
