If from the axioms of Ackermann set theory we remove the second completeness axiom for $V$, and replace it with an axiom of power sets in $V$, that is:
For every element $x$ of $V$: the class of all elements of $V$ that are subclasses of $x$, is an element of $V$.
Formally this is:
$\forall x \in V \ \forall y \ [\forall z (z \in y \leftrightarrow z \subseteq x \wedge z \in V) \to y \in V]$
where: $z \subseteq x \iff \forall m \in z (m \in x)$
Note: the second completeness axiom for $V$ is the axiom:
$\forall x \in V \ \forall y \ (y \subseteq x \to y \in V)$
Question: Is the resulting theory still capable of interpreting ZF?