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

theorem
  for X being Set of U
  for Y being set st Y in X holds Y is Set of U
  proof
    let X be Set of U;
    let Y be set;
    assume
A1: Y in X;
    X is U-set by Def10;
    then Y in union U & U = union U by A1,CLASSES4:81,TARSKI:def 4;
    then Y is U-set;
    hence Y is Set of U by Def10;
  end;
