 Solution

Let $a\in b\in \mathcal{P}(X)$. Since $b\subseteq X$, we have $a\in X$. Transitivity of $X$ gives $a\subseteq X$, so $a\in \mathcal{P}(X)$. This is exactly the membership criterion for a transitive set, proving the power set of a transitive set is transitive.