# 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:

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

$\blacksquare$