> [!NOTE] Axiom (Union) > Given any [[Zermelo Frankel set theory (ZFC)|set]] $a$ of sets, the union of all the members is also a set. That is $\forall a \exists b \forall x [x \in b \leftrightarrow \exists c (c\in a \land x \in c)]$ # Applications **Relation to Pairing**: The axiom of union allows one to unpack a set of sets and thus create a flatter set. Together with the [[Unordered Pair Axiom|axiom of paring]], this implies that for any two sets, there is a set (called their [[Set Union|union]]) that contains exactly the elements of the two sets.