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 $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathbb S \subseteq \mathcal P \left({S}\right)$.

Then:

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

Set of Sets

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

Then:

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

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:

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

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

Proof

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

Similarly for $T$.

$\blacksquare$