Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2007 / B / 6 / ii / Solution

The membership induction principle is the schemaallowing fixed parameters in $\phi$.

Suppose the progressiveness hypothesis holds but $\neg \phi (a)$. The transitive set $T=\mathrm{TC}(\{a\})$ contains $a$. Separation gives the nonempty set $B=\{x\in T:\neg \phi (x)\}$. By Foundation choose $b\in B$ with $b\cap B=\varnothing$. Every $y\in b$ belongs to $T$ because it is a transitive set and cannot belong to $B$, so every such $y$ satisfies $\phi$. Progressiveness now gives $\phi (b)$, a contradiction. Thus the principle holds. The transitive closure is needed to turn a possibly proper class of counterexamples into a set to which Foundation applies.