Cartesian Product is Empty iff Factor is Empty/Family of Sets

Theorem
Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$.

Then:
 * $S = \O$  $S_i = \O$ for some $i \in I$