Given a collection $\mathcal{A}$ of Disjoint nonempty Sets, there exists a set $C$ consisting of exactly one element from each element of $\mathcal{A}$; that is, a set $C$ such that $C$ is contained in the union of the elements of $\mathcal{A}$, and for each $A \in \mathcal{A}$, the set $C \cap A$ contains a single element.

See page 57 of Munkres 2nd ed.

There versions of the axiom which restrict to finite collections