Set is Subset of Union/General Result

Theorem

Let $S$ be a set.

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

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

Then:

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

Proof

Let $x \in T$ for some $T \in \mathbb S$.

Then:

\(\ds x\)

\(\in\)

\(\ds T\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\in\)

\(\ds \bigcup \mathbb S\)

Definition of Set Union

\(\ds \leadsto \ \ \)

\(\ds T\)

\(\subseteq\)

\(\ds \bigcup \mathbb S\)

Definition of Subset

As $T$ was arbitrary, it follows that:

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

$\blacksquare$