Prove that for all $\alpha\in\textbf{On}$ the set $V_{\alpha}$ is transitive

Question

Im a little stuck here. I'm thinking of doing induction on the ordinals $\textbf{On}$, but I can't make it work.

Can someone help me?

Answer 1

Observe $V_\alpha=\bigcup_{\beta<\alpha}\mathscr{P}(V_\beta)$.

Induction: Assume $V_\beta$ is transitive for all $\beta<\alpha$.

Lemma 1. If $X$ is transitive, then $\mathscr{P}(X)$ is transitive.

Lemma 2. The union of transitive sets is transitive.