We work in the set theory NBG with the axiom of (local choice but without global (class) choice. For every class A P(A) is the class of all sets x included in the class A.

We know that P(A) is a set iff A is a set and a proper class iff P(A) is a proper class. We also know that if A is a set there is no bijection between A and P(A), and that P(P(V))=V, where V is the universal class. It is clear that if there is a bijection between A and V, then there is a bijection between A and P(A).

**Question:** Is it true that if there is a bijection between A and P(A) then there is a bijection between A and V?

This question is not interesting under global choice, where all proper classes are bijective.