# 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.

## Also see

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection

- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html