User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-27513631-20161227205129

Sorry, no ZFC can't interpret NBG, but ZFC can interpret NBG + "all classes are definable from sets" (call it NBG*). I don't think I can get a reference, but I can try to outline a proof. (We consider two-sorted NBG* without a separate equality symbol.)

We let the \(V\) of our ZFC model and our NBG* model be isomorphic, and note that the classes of our NBG* model are by definition the closure of \(V\cup\{V\}\) under some (finite) collection of Godel operations. We can then represent each class defined with parameters \(\in P\) as some \(c\in V_\omega[P]\) with an encoding for each godel operation. Then, we can define elementhood between sets as standard, and elementhood between a set and a class by induction on the complexity of the encoding.

I guess that was less precise than I'd have liked it to be.