# Axiom:Axiom of Unions/Set Theory

## Axiom

For every set of sets $A$, there exists a set $x$ (the union set) that contains all and only those elements that belong to at least one of the sets in the $A$:

$\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$

## Also known as

The axiom of unions is in fact most frequently found with the name axiom of union.

However, in some treatments of axiomatic set theory and class theory, for example Morse-Kelley set theory this name is used to mean something different.

Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ specifically uses the plural form axiom of unions for this, and reserves the singular form axiom of union for that.

Some sources refer to the axiom of unions as the axiom of the sum set.