Working in NBG set theory, with AC but without Global Choice, we ask for two proper classes A and B such that A strictly injects in B and B strictly injects in P(A); so
Question: In NBG set theory, is it possible to have two distinct proper classes A and B such that A injects in B and B injects in P(A), but P(A) does not inject in B and B does not i ject in A ?
Gérard Lang



An analogue of CH for proper classes
From the answer given by Ali Enayat to my question ["Bijective equivalent collections of proper classes in set theory" on 14/04/2013](https://mathoverflow.net/questions/124494/bijective-equivalent-collections-of-proper-classes-in-set-theory), we see that there exists a model of NBG where the proper class P(On) is such that:
(i) On injects into P(On), that itself injects into P(P(On))=V, but
(ii) V=P(P(On)) does not inject into P(On) that itself does not inject into On.
So that the answer to the question is positive.
Gérard Lang


