reserve U for Universe;
reserve x for Element of U;

theorem
  for X being Set of U holds U \ X is class of U
  proof
    let X be Set of U;
A1: X is U-set by Def10;
    now
      assume U \ X = {};
      then U c= X by XBOOLE_1:37;
      then X in X by A1;
      hence contradiction;
    end;
    then reconsider UX = U \ X as non empty set;
    X is U-set by Def10;
    then U \ X c= U & not U \ X in U by XBOOLE_1:36,CLASSES4:86;
    then UX is U-Class;
    hence thesis by Def12;
  end;
