Mendelson, in *Introduction to Mathematical Logic*, 4th ed, 1997, had a more elegant approach to comprehension than predecessors, in my opinion.


With $x\in\mathbf{V}$ short for $\exists y(x\in y)$, and $\alpha$ any formula in the language of set theory (possibly without =), use the axiom schemas:


SE: $\exists y(y=\{x|\alpha\})$ and CA: $\forall x(x\in\{x|\alpha\}\leftrightarrow x\in \mathbf{V}\wedge \alpha)$


A class is a set just if it is a member of $\mathbf{V}$. Mendelson goes on and develops NBG set theory on the basis of these, and further assumptions.


*Question:* Have others explored other set theories, with an approach as this?



Have others explored Mendelson's approach to comprehension?
Yes! There are other set theories explored along this way generally speaking.


(1) Quine's Mathematical Logic $\sf ML$ adopted a similar approach on top of his $\sf NF$, and easily one can get a similar treatment on top of $\sf NFU$. See: Quine, Willard Van Orman (1951), Mathematical logic (Revised ed.), Cambridge, Mass.: Harvard University Press, ISBN 0-674-55451-5, MR 0045661


(2) Randall Holmes in his theory about symmetric extensions also used a similar approach. <https://randall-holmes.github.io/Drafts/symmetryrevisited.pdf>


I also used this approach in a closely related article on: <https://arxiv.org/abs/2012.08299>


(3) [Vopenka](https://plato.stanford.edu/entries/settheory-alternative/#VopeAlteSetTheo)'s alternative set theory


(4) Holmes [Pocket](https://plato.stanford.edu/entries/settheory-alternative/#PockSetTheo) set theory


