 Axioms satisfied by hereditarily countable sets

The transitive set of hereditarily countable sets satisfies every ZFC axiom except the Axiom of power set. For internal definable functions on a countable domain, ambient Axiom schema of replacement gives a countable range of hereditarily countable sets, which belongs to the structure by countable-subset closure. The same closure proves axiom schema of collection and retains countable choice-function graphs. All subsets of $\omega$ are elements, but their uncountable full power set is not an element, so Power Set fails.