Union is Smallest Superset/Set of Sets

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Theorem

Let $T$ be a set.

Let $\mathbb S$ be a set of sets.

Then:

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


Proof

By Union of Subsets is Subset: Set of Sets:

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

$\Box$


For the converse implication, 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: Set of Sets 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:

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

$\Box$


Hence:

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

$\blacksquare$


Also see