Union is Smallest Superset

Theorem
Let $S_1$ and $S_2$ be sets.

Then $S_1 \cup S_2$ is the smallest set containing both $S_1$ and $S_2$.

That is:
 * $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \iff \paren {S_1 \cup S_2} \subseteq T$

Necessary Condition
From Union of Subsets is Subset:


 * $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \implies \paren {S_1 \cup S_2} \subseteq T$

Sufficient Condition
Let $\paren {S_1 \cup S_2} \subseteq T$:

Similarly for $S_2$:

That is:


 * $\paren {S_1 \cup S_2} \subseteq T \implies \paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T}$

Thus $\paren {S_1 \subseteq T} \land \paren {S_2 \subseteq T} \iff \paren {S_1 \cup S_2} \subseteq T$ from the definition of equivalence.

Also see

 * Intersection is Largest Subset