Suppose Morse-Kelley set theory consists of class comprehension, class foundation, class extension, axiom of infinity , limitation of size, and the general continuum hypothesis. Can the axiom of powerset,that the power set of a set is a set, be proven in this system? Without gch of course it can’t.

If it can be proven then Morse-Kelley set theory with gch can be presented as class comprehension, class foundation, class extension, axiom of infinity, and the following axiom which combines the axiom of limitation of size and gch: let P be the power class of a class C. If S is a class such that |S|<|P| then there exists an element of P with the same cardinality as S.

Here only one axiom explicitly refers to sets, the axiom of infinity.