Set is Subset of Intersection of Supersets

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

Let $S$ be a subset of both $T_1$ and $T_2$.

Then:
 * $S \subseteq T_1 \cap T_2$

That is:
 * $\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$

Proof 1
Let $S \subseteq T_1 \land S \subseteq T_2$.

Then:

Proof 2
So:
 * $S \subseteq T_1 \land S \subseteq T_2 \implies S \subseteq T_1 \cap T_2$

Also see

 * Union of Subsets is Subset