Axiom:Axiom of Powers/Set Theoretical and Class Theoretical

Axiom of Unions: Difference between Formulations
Recall the two formulations of the axiom of powers:

Formulation 2
Equivalence of Formulations of Axiom of Powers notwithstanding, the two formulations have a subtle difference.

The purely set theoretical (formulation 1) version starts with a given set (of sets), and from it allows the creation of its power set by providing a rule by which this may be done.

The class theoretical (formulation 2) version accepts that such a construct is already constructible in the context of the power set, and is itself a class.

What formulation 2 then goes on to state is that if $x$ is actually a set (of sets), then $\powerset x$ is itself a set.

This is consistent with how:
 * the philosophy of axiomatic set theory defines the constructibility of sets from nothing

differs from
 * the class theoretical approach, in which classes may be considered to be already in existence, and it remains a matter of determining which of these classes are actually sets.