We show how to interpret the language of first-order set
theory in an elementary topos endowed with, as extra structure,
a directed structural system of inclusions (dssi).
As our main result, we obtain
a complete axiomatization of the intuitionistic set theory
validated by all such interpretations.
Since every elementary
topos is equivalent to one carrying a dssi, we thus obtain a
first-order set theory whose
associated categories of sets are exactly the
elementary toposes. In addition, we show that
the full axiom of Separation is validated
whenever the dssi is superdirected. This gives
a uniform explanation for the known facts that cocomplete and realizability
toposes provide models for
Intuitionistic Zermelo—Fraenkel set theory (IZF).