Union of Subsets is Subset/Proof 2

Proof
Let $x \in S_1 \cup S_2$.

By the definition of union, either $x \in S_1$ or $x \in S_2$.

By hypothesis, $S_1 \subseteq T$ and $S_2 \subseteq T$.

By definition of subset:
 * $x \in S_1 \implies x \in T$
 * $x \in S_2 \implies x \in T$

By Proof by Cases it follows that $x \in T$.

Hence the result by definition of subset.