The axiom system consisting of ZF (Zermelo-Fraenkel set theory) along with the axiom of choice ("C"). ZFC is consistent if ZF is (as is ZF + the negation of C). It is sometimes convenient to state that some proof occurs in ZF (i.e. does not require the axiom of choice). There are still some who do not accept C, or have some misgivings about it. And even if you accept C, actually using it in a proof immediately makes the result highly non-constructive. So a proof in ZF is more constructive.