Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2008 / B / 5 / ii

For every $x\in a$, the assumption $a\subseteq V$ supplies an ordinal stage containing $x$. Let ${\alpha }_x$ be the least such ordinal. Leastness makes this a uniquely definable assignment, so ambient replacement gives the setPut $\beta =\sup S$, taking $\beta =0$ if $a$ is empty. Monotonicity of the hierarchy gives $x\in V_{{\alpha }_x}\subseteq V_{\beta }$ for every $x\in a$. Hence $a\subseteq V_{\beta }$, and thereforeThis is a rank-bounding argument: a set-sized collection of elements from the hierarchy has its stages bounded by one ordinal. No simultaneous arbitrary choices are required, since each least stage is uniquely specified.