Set Union Preserves Subsets/Families of Sets

Theorem
Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$.

Let:
 * $\forall \beta \in I: A_\beta \subseteq B_\beta$

Then:
 * $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$

Proof
By definition of subset:
 * $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha \subseteq \bigcup_{\alpha \mathop \in I} B_\alpha$