= Solution
Induct on $\alpha$. For every $\beta<\alpha$, the induction hypothesis says $\beta\in V_{\beta+1}$, and monotonicity gives $V_{\beta+1}\subseteq V_\alpha$. Thus every element of the <ordinal> $\alpha$ lies in $V_\alpha$, so $\alpha\subseteq V_\alpha$. Taking its <power set> gives
$$
\boxed{\alpha\in\mathcal P(V_\alpha)=V_{\alpha+1}.}
$$
For $\alpha=0$ the inclusion is vacuous and $\varnothing\in\mathcal P(\varnothing)$ supplies the base case. This argument uses <ordinal> membership as strict order and the previously proved hierarchy monotonicity.
Back to article page
