Axiom of Choice

Axiom of Choice: For every indexed family $(A_i)_{i\in I}$ of nonempty sets, there exists a function $f:I\to \bigcup_{i\in I} A_i$ such that $f(i)\in A_i$ for all $i\in I$.

Equivalently, every indexed family of sets of nonempty sets admits a choice function selecting one element from each set. In the presence of the other ZF axioms, this axiom is equivalent to Zorn's lemma and to the well-ordering theorem .