nLab
Morse-Kelley set theory

The Morse–Kelley set theory ($\mathrm{MK}$) is an axiomatic approach to set theory which has both classes and sets. Whereas NBG (which also has both classes and sets) is conservative? over ZFC, Morse–Kelley is not a conservative extension of $\mathrm{NBG}$. The principal difference from $\mathrm{NBG}$ is that $\mathrm{MK}$ allows arbitrary formulas $\varphi$ appearing in the class comprehension axiom schema (in particular, formulae with quantifiers ranging over classes themselves).

The approach is explained in the appendix to John Kelley’s 1955 book General Topology.