Possibly something like Separation by $\neg S$, that is:

$\forall a \exists x \forall y (y \in x \leftrightarrow y \in a \land \neg Sy)$.

The idea is that if we define *natural* as: *well founded transitive set of transitive sets, that when non empty then it must have a predecessor, and such that every non empty element of it must have a predecessor*, then this separation would prevent having a natural that fulfills $\neg S$, since otherwise any such a natural $n$ would have a nonempty subset of it that is the set of all of its elements that fulfill $\neg S$, but that set wont have a minimal, thus $n$ won't be well founded which contradicts $n$ being a natural. So this theory would prove that any natural fulfills $S$, therefore the existence of the set of all naturals, which is an inductive set!.

However, I do suspect some kind of inconsistency lurking here or there!