Prove that $\displaystyle A$ is a transitive set if and only if $\displaystyle PA$ is a transitive set.

I think I was able to prove one direction.

Suppose $\displaystyle A$ is transitive. We need to show that $\displaystyle PA \subseteq PPA$. Let $\displaystyle x \in PA$. Then $\displaystyle x \subseteq A$. Since $\displaystyle A$ is transitive, we know $\displaystyle A\subseteq PA$. Hence, $\displaystyle x \subseteq PA \Rightarrow x \in PPA$.

For the other direction. I need to show that $\displaystyle A \subseteq PA$. I have a hard time of effectively using the hypothesis that $\displaystyle PA$ is a transitive set. Can someone give me a hand here? Thanks.