User blog comment:TheKing44/Grand Finale Functions/@comment-35470197-20190611051239/@comment-39605890-20190611060206

NBG and MK are first-order theories of sets and classes. Normally first-order set theory has no classes.

If we interpret NBG and MK as simply being first-order set theories, they would contradict ZFC, since they would not satisfy the axiom of pairing ({V, V} is not a valid object in NBG/MK).