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 \b[the <power set> of a <transitive set> is transitive].

Solution

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

 Solution (source code)