Discussion
In ZFC, the axiom of global choice cannot be stated as such because it involves existential quantification on classes: so it is not a statement of the language of ZFC (nor even an infinite number of statements like axiom schemes requiring universal quantification on classes). It can, however, be stated for a given explicit class, e.g., one can state the fact that such-or-such an explicit class-function is a choice function for V \ { ∅ } or that such-or-such a class-relation is a well-ordering of V: in this form (i.e., for some explicit class function that is tedious but possible to write down), the axiom of global choice follows from the axiom of constructibility.
In Gödel-Bernays, global choice does not add any consequence about sets beyond what could have been deduced from the ordinary axiom of choice.
Global choice is a consequence of the axiom of limitation of size.
Read more about this topic: Axiom Of Global Choice
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