reserve X for set;
reserve UN for Universe;

theorem
  not union rng sequence_univers is Grothendieck
  proof
    set RSU = rng sequence_univers,
        URSU = union RSU;
    assume URSU is Grothendieck;
    then reconsider URSU as non empty Grothendieck;
A1: URSU is axiom_GU1;
    set f = sequence_univers {};
    dom sequence_univers = NAT & rng sequence_univers c= URSU &
      NAT in URSU by A1,Th72,Th94,Def9;
    then URSU in URSU by CLASSES3:def 3;
    hence thesis;
  end;
