# Injection of every proper class in the ordinal class

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 ?

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.