= Solution
The <membership induction theorem> is the following schema, for every formula $\varphi(x)$ with any fixed set parameters:
$$
\boxed{\left[\forall x\left((\forall y\in x\,\varphi(y))\to\varphi(x)\right)\right]
\ \Longrightarrow\ \forall x\,\varphi(x).}
$$
Its hypothesis says the property is progressive under membership. In ZF it follows from the <Axiom of foundation>: if a counterexample $x$ exists, take the nonempty set of counterexamples in the <transitive closure> $\operatorname{TC}(\{x\})$, which contains $x$. A membership-minimal counterexample supplied by the <Axiom of foundation> has no counterexample among its elements, contradicting progressivity. The transitive <closure> ensures that all elements relevant to this argument remain in the set being considered.
Back to article page
