Axiom:Axiom of the Empty Set/Set Theory

Axiom
In the context of axiomatic set theory, the axiom of the empty set is as follows:

There exists a set that has no elements:


 * $\exists x: \forall y: \paren {\neg \paren {y \in x} }$

or:
 * $\exists x: \forall y \in x: y \ne y$

The equivalence of these two formulations is proved by Equivalence of Definitions of Empty Set.

Also see

 * Definition:Zermelo-Fraenkel Axioms
 * Definition:Empty Set
 * Empty Set is Unique
 * Equivalence of Definitions of Empty Set
 * Axiom of Empty Set from Axiom of Infinity and Axiom of Specification