Is it possible in the set theory NBG (with local choice but without global choice) that every proper class injects in the proper class of ordinals ? 



Injection of every proper class in the ordinal class
This is equivalent to global choice, since if $V$ itself injects into Ord, then there is a global well-ordering (defined by $x<y$ if $x$ maps to a smaller ordinal than $y$), and vice versa.


