Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2007 / B / 8 / iii / Solution

Fix $a\in L$. In the ambient universe, Separation forms the setIt is a set of constructible sets, so the previous part gives an ordinal bound on all its members. Increase the bound to a level $L_{\beta }$ also containing $a$. Transitivity ensures $a\subseteq L_{\beta }$. ThenThe inclusion test is absolute for the transitive level $L_{\beta }$, since all members of $x$ and $a$ lie there. The right side is consequently a subset definable over $L_{\beta }$ with parameter $a$, soFor $x\in L$, transitivity of the constructible universe makes internal and ambient subsethood of $a$ agree. Hence $L$ satisfies that $S$ consists exactly of all its subsets of $a$. This is the Power set Axiom in $⟨L,\in ⟩$. It is not the assertion that every ambient subset of $a$ belongs to $L$.