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 $\family {S_i}$.

Proof

Necessary Condition

From Union of Family of Subsets is Subset we have that:

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

$\Box$

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 of Family 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:

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

$\Box$

Hence:

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

$\blacksquare$