User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-27513631-20170122201956

Each NBG class that can be proven to exist is built up from finitely many applications of class operations \(\{P_1,\cdots P_n\}\), for instance those in the wikipedia article, and classes we already have, including sets. In this way, we can construct any class provable to exist in NBG from sets and finitely many operations, which can be encoded into a set \(\ulcorner A\urcorner\) (note that \(\ulcorner\cdot\urcorner\) is not total); we can fairly easily define an accompanying predicate \(\sqin\) by induction such that \(u \sqin \ulcorner A \urcorner \leftrightarrow u \in A\). Note that this includes the full class comprehension schema, hence all classes we can define with only set parameters. Then, we need to assert that \(\forall A\exists b\forall c(c\in A \leftrightarrow c\sqin\ulcorner b\urcorner)\), and we're done.

Also, your second comment is incorrect, due to some of the precise meanings of 'interprets'. For instance, second order arithmetic can interpret ZFC-+(V=HC) but not ZFC-, whereas both of the latter theories can interpret second order arithmetic.

(Here NBG doesn't have Global Choice, as not all ZFC models extend to an NBG*+GC model.)