Union is Smallest Superset/Family of Sets

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

Then for all sets $X$:
 * $\ds \paren {\forall i \in I: S_i \subseteq X} \iff \bigcup_{i \mathop \in I} S_i \subseteq X$

where $\ds \bigcup_{i \mathop \in I} S_i$ is the union of $\family {S_i}$.

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

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

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

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

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

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

So it follows that $S_i \subseteq X$.

So:
 * $\ds \bigcup_{i \mathop \in I} S_i \subseteq X \implies \paren {\forall i \in I: S_i \subseteq X}$

Hence:
 * $\ds \paren {\forall i \in I: S_i \subseteq X} \iff \bigcup_{i \mathop \in I} S_i \subseteq X$

Also see

 * Intersection is Largest Subset: Family of Sets