However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms:[citation needed] Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities.
Together with the axiom schema of replacement, the axiom of union implies that one can form the union of a family of sets indexed by a set.
In the context of set theories which include the axiom of separation, the axiom of union is sometimes stated in a weaker form which only produces a superset of the union of a set.
For example, Kunen[3] states the axiom as which is equivalent to Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions.
(If A is the empty set, then trying to form the intersection of A as is not permitted by the axioms.