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:

$\ds \paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$

Family of Sets

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

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

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

$\ds \paren {\forall X \in \mathbb S: X \subseteq T} \implies \bigcup \mathbb S \subseteq T$

$\Box$

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

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

From Set is Subset of Union: General Result we have that:

$\ds X \subseteq \bigcup \mathbb S$

Thus:

$\ds x \in \bigcup \mathbb S$

But:

$\ds \bigcup \mathbb S \subseteq T$

So it follows that:

$X \subseteq T$

So:

$\ds \bigcup \mathbb S \subseteq T \implies \paren {\forall X \in \mathbb S: X \subseteq T}$

$\Box$

Hence:

$\ds \paren {\forall X \in \mathbb S: X \subseteq T} \iff \bigcup \mathbb S \subseteq T$

$\blacksquare$