Axiom:Axiom of Union

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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.

Then their union $x \cup y$ is a set.