Take all axioms of [Ackermann's set theory][1], remove the class construction axiom schema. 

Add the **limitation of size** axiom:

$\forall X \forall Y [X \in V \land Y \subseteq V \land |Y|\leq |X| \to Y \in V]$

>Would the resulting theory prove $V$ to be a model of $\sf ZF$?


  [1]: https://en.wikipedia.org/wiki/Ackermann_set_theory