Take ZFC, remove axiom of Power set, and put instead of it the following axiom:
Axiom of Successor Cardinals: $\forall \kappa \exists x \forall \alpha ( \alpha \leq \kappa \to \alpha \in x)$
where "$\leq$" refers to "cardinal smaller than or equal" relation, and $\kappa, \alpha$ range over von Neumann ordinals.
Can the resulting theory still interpret ZFC?
The idea is that if we can develop Gödel's constructible universe L inside this system, then this would interpret ZFC? So the power set axiom won't be essential for the development of L?