Textbook Problem

Find the mistake in the following “proof” that for all sets A and B. $\left(A-B\right)\cup \left(A\cap B\right)\subseteq A$.

“Proof: Suppose A and B are any sets, and suppose $x\in \left(A-B\right)\cup \left(A\cap B\right)$.If $x\in A$ then $x\in A-B$ , and so, by definition of difference, $x\in A$ and $x\in B$ . In particular, $x\in A$ , and, therefore, $\left(A-B\right)\cup \left(A\cap B\right)\subseteq A$ by definition of subset.”