It is not a version of set theory at all; there are no specialized set theoretic axioms in CL. CL (without row variables) is simply a generalized form of first-order logic; specifically, it is the result of generalizing the syntax and semantics of standard first-order logic to permit (among other things) self-predication (a.k.a. self-membership (if predicates are taken to denote classes) or self-exemplifation (if predicates are taken to denote properties)). Interested readers might consult
this paper, forthcoming in
Synthese but freely available (for some, at least) online. If you don't have access to the Springer site, a preprint is
available here.