The axiom of union states the following:
If is a (material) set, then there exists a set such that whenever .
Using the axiom of separation (bounded separation is enough), one can prove the existence of a particular set such that the members of the members of are the only members of . Using the axiom of extensionality, we can then prove that this set is unique; it is usually denoted and called the union of (the elements of) .
A slightly different notation may be used when is (Kuratowski)-finite; for example, may be denoted . If is a family of sets, then we may write (and the usual variations for a sequence of sets) for ; however, we require the axiom of replacement to prove that the latter set (the range of the family) exists in general.
If is given as a collection of subsets of some ambient set , then the axiom of union is not necessary; itself already satisfies the conclusion of the hypothesis (and then bounded separation gives us the union that we want). This is the only case when unions are taken in structural set theory. However, structural set theory makes use of disjoint unions, and predicative mathematics requires an axiom giving their existence. (In impredicative mathematics, we can construct disjoint unions from power sets and cartesian products.)