Axiom:Axiom of Choice/Formulation 2

Axiom
Let $\left \langle {X_i} \right \rangle_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.

Then there exists an indexed family $\left \langle {x_i} \right \rangle_{i \mathop \in I}$ such that:
 * $\forall i \in I: x_i \in X_i$

That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.

Also see

 * Equivalence of Versions of Axiom of Choice