
theorem
  for U being set st
  (for x,u being set st x in u in U holds x 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)
  holds U is Grothendieck
  proof
    let U be set;
    assume that
A1: for x,u being set st x in u in U holds x in U and
A2: for x being set st x in U holds bool x in U & union x in U and
A3: omega in U and
A4: 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;
    reconsider U9 = U as non empty set by A3;
    now
      thus U is axiom_GU1 by A1;
      thus U is axiom_GU3 by A2;
      now
        let Y be set;
        let f be Function;
        assume that
A5:     dom f = Y and
A6:     rng f c= U9 and
A7:     Y in U9;
        reconsider f9 = f as Function of Y,rng f by A5,FUNCT_2:1;
        f9 is onto;
        then rng f9 in U9 by A5,A6,A7,A4;
        hence union rng f in U9 by A2;
      end;
      then U9 is FamUnion-closed;
      hence U9 is axiom_GU4 by Th4;
    end;
    hence U is Grothendieck by Th3,Th4,CLASSES3:def 1;
  end;
