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

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?

## marked as duplicate by Andrés E. Caicedo, José Carlos Santos, YuiTo Cheng, Asaf Karagila♦ set-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 12 at 8:15

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