reserve X for set;
reserve UN for Universe;

theorem Th86:
  for X being Element of UN holds not UN \ X is Element of UN
  proof
    let X be Element of UN;
    not UN \ X in UN
    proof
      assume UN \ X in UN;
      then
A1:   (UN \ X) \/ X in UN by CLASSES2:60;
A2:   (UN \ X) \/ X = UN \/ X by XBOOLE_1:39;
A3:   UN \/ X c= UN
      proof
        let x be object;
        assume x in UN \/ X;
        then x in UN or (x in X in UN) & UN is axiom_GU1 by XBOOLE_0:def 3;
        hence thesis;
      end;
      UN = UN \/ X by A3,XBOOLE_1:7;
      hence thesis by A1,A2;
    end;
    hence thesis;
  end;
