Set Union Preserves Subsets/Proof 2

Theorem

Let $A, B, S, T$ be sets.

Then:

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

Proof

By Subset Relation is Transitive, $\subseteq$ is a transitive relation.

By the corollary to Set Union Preserves Subsets (Proof 2), $\subseteq$ is compatible with $\cup$.

Thus the theorem holds by Operating on Transitive Relationships Compatible with Operation.

$\blacksquare$