ZF

1971, Ulrich Felgner, Models of ZF-Set Theory, Springer, Lecture Notes in Mathematics 223, page 21, 1. Corollary: ZF is not finitely axiomatizable. 2. Corollary: ZF is reflexive (i.e. the consistency of every finite subtheory of ZF can be proved within ZF).

1991 [Kluwer Academic], Fred Landman, Structures for Semantics, 1991, Springer, Softcover, page 56, However, the problem with it, and the reason why it is not part of ZF strictly (apart from the fact that it implies the axiom of choice) is that it is rather arbitrary.