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



Injection of the proper class of ordinals in every proper class
The answer is no. That principle is equivalent to global choice.


To see this, consider the class $W$ consisting of all
well-orderings of any rank-initial segment $V\_\alpha$, for any
$\alpha$. If we had an injection of Ord into $W$, then there must
be unboundedly many $\alpha$'s that are used, since each
$V\_\alpha$ has only a set-sized family of well-orderings. Thus, we
have a global selection of well-orderings of unboundedly many
$V\_\alpha$, and from this we can define a well-ordering of the
entire universe. Namely, $x<y$ if the rank of $x$ is lower than
that of $y$, or if they have the same rank and $x<y$ in the first
well-ordering of some sufficiently large $V\_\alpha$ to appear in the range of the
injection of Ord into $W$.


**Update.** I made a blog post concerning these various equivalent formulations of [The global choice principle in Gödel-Bernays set theory](http://jdh.hamkins.org/the-global-choice-principle-in-godel-bernays-set-theory/), in which I explain this answer and give several other related formulations and arguments. 


