Union is Smallest Superset/Family of Sets

Theorem
Let $\left \langle{S_i}\right \rangle_{i \mathop \in I}$ be a family of sets indexed by $I$.

Then for all sets $X$:
 * $\displaystyle \left({\forall i \in I: S_i \subseteq X}\right) \iff \bigcup_{i \mathop \in I} S_i \subseteq X$

where $\displaystyle \bigcup_{i \mathop \in I} S_i$ is the union of $\left \langle{S_i}\right \rangle$.

Necessary Condition
From Union of Subsets is Subset/Family of Sets we have that:
 * $\displaystyle \left({\forall i \in I: S_i \subseteq X}\right) \implies \bigcup_{i \mathop \in I} S_i \subseteq X$

Sufficient Condition
Now suppose that $\displaystyle \bigcup_{i \mathop \in I} S_i \subseteq X$.

Consider any $i \in I$ and take any $x \in S_i$.

From Set is Subset of Union: Family of Sets we have that:
 * $\displaystyle S_i \subseteq \bigcup_{i \mathop \in I} S_i$

Thus:
 * $\displaystyle x \in \bigcup_{i \mathop \in I} S_i$

But:
 * $\displaystyle \bigcup_{i \mathop \in I} S_i \subseteq X$

So it follows that $S_i \subseteq X$.

So:
 * $\displaystyle \bigcup_{i \mathop \in I} S_i \subseteq X \implies \left({\forall i \in I: S_i \subseteq X}\right)$

Hence:
 * $\displaystyle \left({\forall i \in I: S_i \subseteq X}\right) \iff \bigcup_{i \mathop \in I} S_i \subseteq X$

Also see

 * Intersection is Largest Subset: Family of Sets