
theorem
  for X being set holds Universe_closure {X} = GrothendieckUniverse X
  proof
    let X be set;
    set UC = Universe_closure {X},
        GU = GrothendieckUniverse X;
    reconsider UC as Grothendieck;
A1: {X} c= Universe_closure {X} &
    for Y be Universe st {X} c= Y holds Universe_closure {X} c= Y
      by CLASSES2:def 4;
A2: for X,Y be set holds {X} c= Y iff X in Y
    proof
      let X,Y be set;
      hereby
        assume
A3:     {X} c= Y;
        X in {X} by TARSKI:def 1;
        hence X in Y by A3;
      end;
      assume X in Y;
      hence thesis by TARSKI:def 1;
    end;
    then reconsider UC as Grothendieck of X by A1,CLASSES3:def 4;
    for Y be Universe st X in Y holds Universe_closure {X} c= Y by A1,A2;
    then Universe_closure {X} c= GrothendieckUniverse X & GU c= UC
      by CLASSES3:def 4,def 5;
    hence thesis;
  end;
