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Blueberry comments on But Somebody Would Have Noticed - Less Wrong
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Yes, and I'm guessing cousin_it's version of set theory is possibly equivalent to something similar. I'd love to hear more about it.
Well I mean, I imagine it shouldn't be too hard to take ZFA (or similar) and tack proper classes onto it. Logic is not really my thing so I'm not actually familiar with how you show that NBG conservatively extends ZFC. The result would be a bit odd, though, in that classes would act very differently from sets - well, OK, more differently than they already do in NBG...
I don't know the proof either. The other weird thing to note is that even though NBG is a conservative extension of ZFC, some proofs in NBG are much shorter than proofs in ZFC. So in some sense it is only weakly conservative. I don't know if that notion can be made at all more precise.
Edit: Followup thought, most interesting conservative extensions are only weakly conservative in some sense. Consider for example finite degree field extensions of Q. If axiomatized these become conservative extensions of Z. (That's essentially why for example we can prove something in the Gaussian integers and know there's a proof in Z).