# 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:

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

$\blacksquare$