User blog comment:Rgetar/Designation for the next element of a class above an ordinal and some rules for booster-base/@comment-35470197-20200219005322/@comment-35470197-20200220102631

> I guess the first option is correct, since to prove or disprove it one need to consider "a collection of classes", but it is invalid in ZFC

Right. When we deal with a "collection" of classes, we usually use a meta-theoretic argument on a set of (Goedel codes of) formulae, which is outside of ZFC set theory.