# Set is Subset of Union

## Theorem

The union of two sets is a superset of each:

$S \subseteq S \cup T$
$T \subseteq S \cup T$

### General Result

Let $S$ be a set.

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

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

Then:

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

### Set of Sets

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

Then:

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

### Indexed Family of Sets

In the context of a family of sets, the result can be presented as follows:

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Then:

$\ds \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$

where $\ds \bigcup_{\alpha \mathop \in I} S_\alpha$ is the union of $\family {S_\alpha}$.

## Proof

 $\ds x \in S$ $\leadsto$ $\ds x \in S \lor x \in T$ Rule of Addition $\ds$ $\leadsto$ $\ds x \in S \cup T$ Definition of Set Union $\ds$ $\leadsto$ $\ds S \subseteq S \cup T$ Definition of Subset

Similarly for $T$.

$\blacksquare$