User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-27513631-20170125001457

ZFC- (usually written \(ZFC^-\))is ZFC without power set (but with Collection instead of Replacement, as that is stronger in the absence of Power Set).

By class operations, I mean some collection of terms \(t_1(A),\cdots t_n(A,B)\), which are functions from classes to classes. They have to be defined directly, as we can't have classes of classes.

And I think we may be talking about the same thing, as far as interpretability goes? I agreed with everything the MSE comment said, and only learnt that what I thought was bi-interpretable is actually mutually interpretable, which is slightly weaker. Is the \(ZFC^-\) vs \(ZFC^{-\inf}\) what made us misunderstand each other?