reserve X, Y, Z for set, x, y, z for object;

theorem
  (for x holds not x in X iff (x in Y iff x in Z)) implies X = Y \+\ Z
proof
  assume
A1: not x in X iff (x in Y iff x in Z);
  now
    let x;
    x in X iff x in Y & not x in Z or x in Z & not x in Y by A1;
    then x in X iff x in Y \ Z or x in Z \ Y by Def5;
    hence x in X iff x in Y \+\ Z by Def3;
  end;
  hence thesis by TARSKI:2;
end;
