reserve X for set;
reserve UN for Universe;

theorem
  not the set of all u where u is Element of UN is Element of UN
  proof
    set SETUN = the set of all u where u is Element of UN;
    assume
A1: SETUN is Element of UN;
    now
      thus SETUN c= UN
      proof
        let o be object;
        assume o in SETUN;
        then ex u be Element of UN st o = u;
        hence o in UN;
      end;
      thus UN c= SETUN;
    end;
    then SETUN = UN;
    then UN in UN by A1;
    hence thesis;
  end;
