Past exam of the logic and set theory course of the University of Oxford / 2011 / B / 5 / a

A transitive set $X$ is one for which every member is a subset: $\forall y\in X(y\subseteq X)$, equivalently $\bigcup X\subseteq X$. If $x\in y\in X$, the subset condition puts $x\in X$. Conversely, if this membership implication always holds, every element of each $y\in X$ belongs to $X$, so $y\subseteq X$. Thus

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.