AC may be incorporated into the subset axiom.
The subset axiom says that, assuming the use of "vacuous truth," any set X has a subset.
AC would say that every pure set except those of cardinality less than 2 (as in {0,1}) has a nonempty proper subset.
But such a formulation requires an axiom of foundation, or the equivalent, in order to outlaw sets of "cardinality" less than 0. (That is, to forbid infinite descent definitions of not-so-primitive elements.)
