Set Union Preserves Subsets/Proof 2
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Let $A, B, S, T$ be sets.
- $A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$
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.