Power-set axiom at limit ranks
= Power-set axiom at limit ranks
{title2=$(V_\delta,\in)\models\mathrm{PowerSet}$}
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.