# Set is Subset of Intersection of Supersets/Proof 1

## 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

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

Then:

 $\ds x \in S$ $\leadsto$ $\ds x \in T_1 \land x \in T_2$ Definition of Subset $\ds$ $\leadsto$ $\ds x \in T_1 \cap T_2$ Definition of Set Intersection $\ds$ $\leadsto$ $\ds S \subseteq T_1 \cap T_2$ Definition of Subset