The proof that axiom of foundation implies Epsilon induction in $ZF$ that I know of relies on existence of Transitive closures for all sets, but Zermelo set theory doesn't prove those, so is it the case that axiom of foundation also implies Epsilon induction given just the axioms of Zermelo $``Z"$? i.e. in a milieu that lacks the necessary apparatus for defining transitive closures for all sets. So is it consistent with $Z$ to have a set $s$ and define predicate $\varphi$ in the langauge of $Z$ such that it is provable to have $\forall y [\forall x \in y (\varphi(x)) \to \varphi(y)] $ and at the same time have $\neg \varphi(s)$? of course $s$ must not have a transitive closure, and this is known to be consistent with $Z$.
By $Z$ here it is meant the version of $Z$ in first order logic with identity and membership that is axiomatized by axioms of identity and: Extensionality, Foundation, Separation, Empty set, Pairing, Set union, Power, Infinity. Those are present here.(But add to it the axiom of foundation as every non-empty set must have an element that is disjoint from it as present here)
But without including axiom of Choice.