# Axiom:Axiom of Unions

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## Axiom

### Set Theory

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 $A$:

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

### Class Theory

Let $x$ be a set (of sets).

Then its union $\displaystyle \bigcup x$ is also a set.

## 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**.