# Axiom:Axiom of Union/Zermelo-Fraenkel

## Axiom

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

- $\forall A: \exists x: \forall y: \left({y \in x \iff \exists z: \left({z \in A \land y \in z}\right)}\right)$

## Also known as

Some sources refer to this as the **Axiom of the Sum Set**.

Some give this a plural: **Axiom of Unions**.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 4$: Unions and Intersections

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