
theorem
  for U being Grothendieck st omega in U holds
  (for x,u being set st x in u in U holds x in U) &
  (for u,v being set st u in U & v in U holds {u,v} in U & [u,v] in U &
                                              [:u,v:] in U) &
  (for x being set st x in U holds bool x in U & union x in U) &
  (omega in U) &
  (for a,b being set for f being Function of a,b st dom f = a &
  f is onto & a in U & b c= U holds b in U)
  proof
    let U be Grothendieck;
    assume
A1: omega in U;
    then reconsider G = U as non empty Universe;
    hereby
      let x,u be set;
      assume
A2:   x in u in U;
      U is axiom_GU1;
      hence x in U by A2;
    end;
    hereby
      let u,v be set;
      assume that
A3:   u in U and
A4:   v in U;
      reconsider u9 = u,v9 = v as Element of G by A3,A4;
      {u9,v9} is Element of G & [u9,v9] is Element of G &
        [:u9,v9:] is Element of G;
      hence {u,v} in U & [u,v] in U & [:u,v:] in U;
    end;
    hereby
      let x be set;
      assume x in U;
      then reconsider x9 = x as Element of G;
      bool x9 in G & union x9 in G;
      hence bool x in U & union x in U;
    end;
    thus omega in U by A1;
    thus for a,b be set for f be Function of a,b st dom f = a &
      f is onto & a in U & b c= U holds b in U by CLASSES3:2;
  end;
