Intersection is Largest Subset

Theorem
Let $T_1$ and $T_2$ be sets.

Then $T_1 \cap T_2$ is the largest set contained in both $T_1$ and $T_2$.

That is:
 * $S \subseteq T_1 \land S \subseteq T_2 \iff S \subseteq T_1 \cap T_2$

Proof
Let $S \subseteq T_1 \land S \subseteq T_2$.

Then:

Alternatively:

So:
 * $S \subseteq T_1 \land S \subseteq T_2 \implies S \subseteq T_1 \cap T_2$.

Now let $S \subseteq T_1 \cap T_2$.

From Intersection is Subset we have $T_1 \cap T_2 \subseteq T_1$ and $T_1 \cap T_2\subseteq T_2$.

From Subset Relation is Transitive, it follows directly that $S \subseteq T_1$ and $S \subseteq T_2$.

So $S \subseteq T_1 \cap T_2 \implies S \subseteq T_1 \land S \subseteq T_2$.

From the above, we have:


 * 1) $S \subseteq T_1 \land S \subseteq T_2 \implies S \subseteq T_1 \cap T_2$;
 * 2) $S \subseteq T_1 \cap T_2 \implies S \subseteq T_1 \land S \subseteq T_2$.

Thus $S \subseteq T_1 \land S \subseteq T_2 \iff S \subseteq T_1 \cap T_2$ from the definition of equivalence.

Also see

 * Union Smallest