This question asks about the Set Theory summarized in the appendix of Kelley's book General Topology.
Is it equivalent to some of the MK, NBG or ZFC set theories? Or is it different (maybe partial equivalent, ''embedded'')?
Direct link to page: https://archive.org/stream/GeneralTopology/Kelley-GeneralTopology#page/n267/mode/2up
The system of axioms adopted is a variant of systems of Skolem and of A. P. Morse and owes much to the Hilbert-Bernays-von Neumann system as formulated by Godel. The formulation used here is designed to give quickly and naturally a foundation for mathematics which is free from the more obvious paradoxes.
This system is exactly MK. Indeed, MK stands for Morse-Kelley, and this appendix is one of the places that MK set theory was originally formulated (and is the reason it is named after Kelley).