Cartesian Product of Subsets/Family of Nonempty Subsets

Theorem
Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets where $I$ is the indexing set.

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

Let $\family {T_i}_{i \mathop \in I}$ be another family of sets.

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

Let $T_i \neq \O$ for all $i \in I$.

Then:
 * $T \subseteq S \iff \forall i \in I : T_i \subseteq S_i$.