= Solution
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 $u\in v\in\bigcup_iT_i$, choose $i$ with $v\in T_i$ and infer $u\in T_i$. This proves the limit step and hence \b[every $V_\alpha$ is transitive].
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