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