Axiom:Axiom of 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 Subsets and some treatments exclude it from the list.