Set Union Preserves Subsets/Corollary/Proof 2

Proof
Let $A$, $B$, and $S$ be sets.

Let $A \subseteq B$.

Let $x \in A \cup S$.

By the definition of union:
 * $x \in A$ or $x \in S$

Suppose $x \in A$.

Then by the definition of subset:
 * $x \in B$

Thus by the definition of union:
 * $x \in B \cup S$

Suppose instead that $x \in S$.

Then by the definition of union:
 * $x \in B \cup S$

Thus for all $x \in A \cup S$:
 * $x \in B \cup S$

The second result follows from Union is Commutative.