reserve X for set;
reserve UN for Universe;

theorem
  for UN1,UN2 being Universe
  for a being Element of UN1 st not a in UN2 holds UN2 in UN1
  proof
    let G1,G2 be Universe;
    let a be Element of G1;
    assume
A1: not a in G2;
A2: G1 is Grothendieck of {a} by CLASSES3:def 4;
A3: now
      assume {a} meets G2;
      then {a} /\ G2 is non empty;
      then consider z be object such that
A4:   z in {a} /\ G2;
      z in {a} & z in G2 by A4,XBOOLE_0:def 4;
      hence contradiction by A1,TARSKI:def 1;
    end;
    now
      assume
A5:   G1 in G2;
      G2 is axiom_GU1 & G2 is axiom_GU3;
      hence contradiction by A1,A5;
    end;
    hence thesis by A3,A2,Th78,CLASSES2:52;
  end;
