The Axiom of Choice: For any family $A$ of nonempty, disjoint sets, there exists a set that consists of exactly one element from each element of $A$.

The Choice-Function Principle: For any family $A$ of nonempty sets, there exists a function $f:A \rightarrow \cup_{a \in A} a$ such that for every $a \in A$, $f(a) \in A$