Power-set axiom at limit ranks ($(V_{\delta },\in )\models \mathrm{PowerSet}$)

For a nonzero limit ordinal $\delta$, the structure $(V_{\delta },\in )$ satisfies the Axiom of power set. For $x\in V_{\beta }$ with $\beta <\delta$, transitivity gives $x\subseteq V_{\beta }$, so $\mathcal{P}(x)\subseteq V_{\beta +1}$ and $\mathcal{P}(x)\in V_{\beta +2}\subseteq V_{\delta }$. Every external subset is therefore present and the external power set is also the internal one.