Is limitation of size provable in NBG?

It seems that certain axioms of NBG are implied by limitation of size, allowing one to omit them for a shorter axiom list when assuming limitation of size. But is limitation of size provable in standard NBG with global choice? If so, can someone point me to a proof?

Solution 1:

Global choice is limitation of size. The reason is simple. Global choice implies that there is a bijection between $V$ and the class of ordinals. Simply choose for each $V_\alpha$ a well-ordering, and glue them together.

Now that there is a bijection between $V$ and the ordinals, every class, being a subclass of $V$, will have a bijection with a subclass of the ordinals. But, lo and behold, every proper subclass of the ordinals is bijective with the ordinals again.

Therefore, every two classes are the same size, assuming global choice.