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.



When do we have a bijection between a proper class A and its power set class P(A)?
Yes, this is provable in NBG. To see this, let $F$ be a one-one function from $\mathcal P(A)$ into $A$. By transfinite recursion on $\in$, we define a function $G$ from $V$ to $A$ such that $G(x) = F(G[x])$. A simple induction then establishes that $G$ is one-one.


