When do we have a bijection between a proper class A and its power set class P(A)? 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.
 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.
A: It suffices to show that for $A=\sf Ord$. Since in that case global choice holds and the rest follows.
Fix a pairing function for ordinals, namely a bijection between $\sf Ord$ and $\sf Ord\times Ord$. Now given a bijection between $\sf Ord$ and $\cal P\sf (Ord)$, pick for each $\alpha$ the least $X_\alpha\in\cal P\sf (Ord)$ such that $X_\alpha$ encodes the structure $(V_\alpha,\in)$, and $X_\alpha$ is disjoint from $\bigcup_{\beta<\alpha} X_\beta$. Namely a subset of the ordinals which, when translated using the pairing function, gives us a structure which is extensional and well-founded, and its transitive collapse is $V_\alpha$.
Now we have a bijection between the ordinals and the universe, defined by the following, given $x$ of rank $\alpha$, $x$ is mapped to the unique $\eta\in X_\alpha$ which encodes $x$.
(Alternatively, you don't need to require that the $X_\alpha$'s are disjoint, and this defines a surjection from $\sf Ord$ onto $V$ in a similar way.)
