Axiom:Axiom of the Empty Set

Axiom
There exists a set that has no elements:


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

Also defined as
It can equivalently be specified:
 * $\exists x: \forall y \in x: y \ne y$

The equivalence is proved by Equivalence of Definitions of Empty Set.

Also known as
This axiom is also known as the Axiom of Existence, but there exists another axiom with such a name.

Hence it is preferable not to use that name.

Relation to other axioms
This can be deduced from the Axiom of Infinity and axiom of specification and some treatments exclude it from the list.

Also see

 * Axiom:Zermelo-Fraenkel Axioms
 * Definition:Empty Set
 * Empty Set is Unique