theorem Th4:
  x in succ X iff x in X or x = X
proof
  thus x in succ X implies x in X or x = X
  proof
    assume x in succ X;
    then x in X or x in { X } by XBOOLE_0:def 3;
    hence thesis by TARSKI:def 1;
  end;
  assume x in X or x = X;
  then x in X or x in { X } by TARSKI:def 1;
  hence thesis by XBOOLE_0:def 3;
end;
