Set Union Preserves Subsets/Corollary/Proof 1

 Let $A, B, S$ be sets.

Then:
 * $A \subseteq B \implies A \cup S \subseteq B \cup S$
 * $A \subseteq B \implies S \cup A \subseteq S \cup B$

Proof
Let $A \subseteq B$, and let $S$ be any set.

From Set Union Preserves Subsets, substituting $S$ for $T$:


 * $A \subseteq B, \ S \subseteq S \implies A \cup S \subseteq B \cup S$

From Subset of Itself, $S \subseteq S$ for all sets $S$.

Hence the first result:
 * $A \subseteq B \implies A \cup S \subseteq B \cup S$

The second result follows from Union is Commutative.