In the formal language of the Zermelo-Frankel axioms, the axiom reads:
The axiom of empty set may also be seen as a special case of a generalisation of the axiom of pairing.
In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of infinity. On the other hand, there are other formulations of that axiom that don't presuppose the existence of an empty set. Also, the ZF axioms can also written using a constant predicate representing the empty set; then the axiom of infinity uses this predicate without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty. Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set will still be required. That said, any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation.