
theorem Th3:
  for X being set holds X is axiom_GU1 iff X is epsilon-transitive
  proof
    let X be set;
    hereby
      assume X is axiom_GU1;
      then for x be set st x in X holds x c= X;
      hence X is epsilon-transitive;
    end;
    assume
A1: X is epsilon-transitive;
    now
      let x,y be set;
      assume that
A2:   y in x and
A3:   x in X;
      x c= X by A3,A1;
      hence y in X by A2;
    end;
    hence thesis;
  end;
