Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 4 / ii / a

A set $T$ is transitive if every element of an element of $T$ is itself in $T$, equivalently each element of $T$ is a subset of $T$. Proceed by transfinite induction. The empty set is transitive. If $T$ is transitive and $u\in v\in \mathcal{P}(T)$, then $v\subseteq T$ gives $u\in T$, and transitivity of $T$ gives $u\subseteq T$. Thus $u\in \mathcal{P}(T)$, proving the successor stage transitive. A union of transitive sets is transitive: if $u\in v\in {\bigcup }_i{T}_i$, choose $i$ with $v\in T_i$ and infer $u\in T_i$. This proves the limit step and hence every $V_{\alpha }$ is transitive.