Union is Smallest Superset/General Result

Theorem
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S$ be a subset of $\mathcal P \left({S}\right)$.

Then:
 * $\displaystyle \left({\forall X \in \mathbb S: X \subseteq T}\right) \iff \bigcup \mathbb S \subseteq T$

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

Proof
Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Suppose that $\forall X \in \mathbb S: X \subseteq T$.

Consider any $\displaystyle x \in \bigcup \mathbb S$.

By definition of set union, it follows that:
 * $\exists X \in \mathbb S: x \in X$

But as $X \subseteq T$ it follows that $x \in T$.

Thus it follows that:
 * $\displaystyle \bigcup \mathbb S \subseteq T$

So:
 * $\displaystyle \left({\forall X \in \mathbb S: X \subseteq T}\right) \implies \bigcup \mathbb S \subseteq T$

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

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

From Subset of Union: General Result we have that $X \subseteq \bigcup \mathbb S$.

Thus $\displaystyle x \in \bigcup \mathbb S$.

But $\displaystyle \bigcup \mathbb S \subseteq T$.

So it follows that $X \subseteq T$.

So:
 * $\displaystyle \bigcup \mathbb S \subseteq T \implies \left({\forall X \in \mathbb S: X \subseteq T}\right)$

Hence:
 * $\displaystyle \left({\forall X \in \mathbb S: X \subseteq T}\right) \iff \bigcup \mathbb S \subseteq T$

Also see

 * Intersection Largest/General Result