reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
 for x being set holds
  x is U-petit iff card x in card U
  proof
    let x be set;
    hereby
      assume x is U-petit;
      then consider u be Element of U such that
A1:   u,x are_equipotent;
      card u = card x by A1,CARD_1:5;
      hence card x in card U by CLASSES4:30;
    end;
    assume card x in card U;
    then reconsider u = card x as Element of U by CLASSES2:13;
    card u = card x;
    hence x is U-petit by CARD_1:5;
  end;
