
theorem
  for X being set holds X \+\ {} = X
proof
  let X be set;
  thus X \+\ {} c= X
  proof
    let x be object;
    assume x in X \+\ {};
    then
A1: x in X \ {} or x in {} \ X by XBOOLE_0:def 3;
    per cases by A1,XBOOLE_0:def 5;
    suppose
      x in X & not x in {};
      hence thesis;
    end;
    suppose
      x in {} & not x in X;
      hence thesis by XBOOLE_0:def 1;
    end;
  end;
  let x be object;
A2: not x in {} by XBOOLE_0:def 1;
  assume x in X;
  then x in X \ {} by A2,XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 3;
end;
