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
From the answer given by Ali Enayat to my question "Bijective equivalent collections of proper classes in set theory" on 14/04/2013, 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

$\begingroup$ It would help to give the answer in the form A=..., B=.... $\endgroup$ – Matt F. Sep 10 '17 at 18:17