Set is Subset of Union

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

Proof
$$x \in S$$

$$\Leftrightarrow x \in S \lor x \in T$$ (Rule of Addition)

$$\Leftrightarrow x \in \left({S \cup T}\right)$$ (Definition of Set Union)

$$\Leftrightarrow S \subseteq S \cup T$$ (Subset definition)