Prove or disprove that $\displaystyle A\setminus(A \setminus B) \subseteq B$

$\displaystyle \textbf{Proof. }$ Let $\displaystyle x \in A \setminus (A \setminus B)$. Then $\displaystyle x \in A $ and $\displaystyle x \notin A \setminus B $. Since $\displaystyle x \notin A \setminus B$, $\displaystyle x \notin A$ or $\displaystyle x \in B$. Since it can't be the case that $\displaystyle x \notin A$, we must have that $\displaystyle x \in B$. Therefore, $\displaystyle A \setminus (A \setminus B) \subseteq B$

The proof seems to make sense (to me at least), but what if $\displaystyle A$ and $\displaystyle B$ are disjoint? Wouldn't that mean that $\displaystyle A \setminus (A\setminus B) = \emptyset$?