# Axiom:Axiom of Union

## Axiom

### Zermelo-Fraenkel Set Theory

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:

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

### Morse-Kelley Set Theory

Let $x$ and $y$ be sets.