Past exam of the axiomatic set theory course of the University of Oxford / 2011 / 1 / b / Solution

Fix $x\in V_{\delta }$. Since $\delta$ is a limit ordinal, choose $\beta <\delta$ with $x\in V_{\beta }$. Transitivity gives $x\subseteq V_{\beta }$. Every subset $u\subseteq x$ therefore belongs to $V_{\beta +1}=\mathcal{P}(V_{\beta })$, and consequentlyThe last inclusion uses $\beta +2<\delta$, since a finite number of successor steps remains below a limit ordinal. In particular every subset of $x$ is also an element of the transitive set $V_{\delta }$.

Take $y=\mathcal{P}(x)$. For any $u\in V_{\delta }$, transitivity ensures that the internal statement $u\subseteq x$ agrees with the external one. HenceSuch a $y\in V_{\delta }$ exists for every $x\in V_{\delta }$, so $(V_{\delta },\in )$ satisfies the Axiom of power set. This power-set axiom at limit ranks actually holds for every nonzero limit ordinal; the stated restriction $\delta >\omega$ is sufficient.