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