Set is Subset of Union/Set of Sets

Theorem

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

Then:

$\ds \forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$

Proof

Let $T$ be any element of $\mathbb S$.

We wish to show that $T \subseteq S$.

Let $x \in T$.

Then:

\(\ds x\)

\(\in\)

\(\ds T\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\in\)

\(\ds \bigcup \mathbb S\)

Definition of Set Union

Since this holds for each $x \in T$:

\(\ds T\)

\(\subseteq\)

\(\ds \bigcup \mathbb S\)

Definition of Subset

As $T$ was arbitrary, it follows that:

$\forall T \in \mathbb S: T \subseteq \bigcup \mathbb S$

$\blacksquare$