Classes are often informally thought of as being "larger" than sets. Usually, the notion of "larger" is formalized via an injection: $B$ is "at least as large" as $A$ iff there is an injection from $A$ to $B$, and strictly "larger" if there is no injection going the other way.
Even though ZFC does not formalize the notion of proper classes, we can still speak sensibly of the notion of an injection from a set to a class, which is simply a function such that its image is within the class. So we can still say a set injects into some proper class if the relevant function exists.
It is known that without AC, we can have sets that are "incomparable" in size, because the universe is now only partially ordered rather than well-ordered. But it seems that we must now also have the situation where sets can be incomparable in size to classes!
Otherwise, if every set injected into every class, they would all inject into the ordinals, and hence be well-ordered. (Right?)
So without AC, we cannot say that classes are necessarily "larger" than sets -- just "different"!
Is this the right understanding?
Is the axiom "every set injects into every class" equivalent to the axiom of choice?
Is there some sensible way to interpret the notion of "class" other than as an entity too "large" to be a set, since this makes no sense without AC?
I would be happy to discuss within a theory like NBG as well that explicitly formalizes classes, although I tried to word this in such a way that it wasn't necessary.