Union is Smallest Superset/General Result

Theorem
Let $S$ and $T$ be sets.

Let $\powerset S$ denote the power set of $S$.

Let $\mathbb S$ be a subset of $\powerset S$.

Then:
 * $\displaystyle \paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$

Family of Sets
In the context of a family of sets, the result can be presented as follows:

Proof
Let $\mathbb S \subseteq \powerset S$.

By Union of Subsets is Subset: Subset of Power Set:
 * $\displaystyle \paren {\forall X \in \mathbb S: X \subseteq T} \implies \bigcup \mathbb S \subseteq T$

Now suppose that $\displaystyle \bigcup \mathbb S \subseteq T$.

Consider any $X \in \mathbb S$ and take any $x \in X$.

From Set is Subset of Union: General Result we have that:
 * $\displaystyle X \subseteq \bigcup \mathbb S$

Thus:
 * $\displaystyle x \in \bigcup \mathbb S$

But:
 * $\displaystyle \bigcup \mathbb S \subseteq T$

So it follows that:
 * $X \subseteq T$

So:
 * $\displaystyle \bigcup \mathbb S \subseteq T \implies \paren {\forall X \in \mathbb S: X \subseteq T}$

Hence:
 * $\displaystyle \paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$

Also see

 * Intersection is Largest Subset