My understanding is that ZF posits the existence of just one set, the empty set $\emptyset$, and the von Neumann hierarchy is constructed starting from there using the axioms of power set and infinity.
The axiom of regularity $$ \forall x (x \neq \emptyset \implies \exists y (y \in x \land \forall z (z \in x \implies z \notin y)))$$ asserts the existence of an $\epsilon$-minimal member $y$ of every non-empty set $x$.
Can one prove that every set in the von Neumann hierarchy has an $\epsilon$-minimal member? If so then why do we need the axiom of regularity? Or is the von Neumann hierarchy just some sort of "minimal model" which satisfies the ZF axioms and the whole point is to prove theorems about other models which you don't know a priori satisfy regularity?