A weaker Axiom of Infinity?
Your axiom is actually equivalent to the Axiom of Infinity modulo ZFC-{Infinity}. This is very easy to see via the replacement axiom, but it follows even just using Extensionality, Pairs, Unions, Specification and Powers: you can prove a version of the recursion theorem, from which you can extract the usual set $\omega$ of von Neumann numerals.