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This question already has an answer here:

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?

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marked as duplicate by Andrés E. Caicedo, José Carlos Santos, YuiTo Cheng, Asaf Karagila set-theory Jun 12 at 8:15

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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.

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    $\begingroup$ ah okay, i think i got it. And $V_0=\emptyset$, sot its trivail? for the indtuction start? $\endgroup$ – Esteban Cambiasso Jun 12 at 7:53

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