Intersection of Transitive Relations is Transitive

Theorem
The intersection of two transitive relations is also a transitive relation.

General Statement
Let $\left\{{\mathcal{R}_i: i \in I}\right\}$ be an $I$-indexed collection of transitive relations on a set $S$.

Then their intersection $\displaystyle \bigcap_{i \in I} \mathcal{R}_i$ is also a transitive relation on $S$.

Proof
Let $\mathcal R_1$ and $\mathcal R_2$ be transitive relations (on what sets is immaterial for this argument).

Let $\left({s_1, s_2}\right) \in \mathcal R_1 \cap \mathcal R_2$ and $\left({s_2, s_3}\right) \in \mathcal R_1 \cap \mathcal R_2$.

Then by definition of intersection:
 * $\left({s_1, s_2}\right) \in \mathcal R_1$ and $\left({s_1, s_2}\right) \in \mathcal R_2$
 * $\left({s_2, s_3}\right) \in \mathcal R_1$ and $\left({s_2, s_3}\right) \in \mathcal R_2$

Then as $\mathcal R_1$ and $\mathcal R_2$ are both transitive:
 * $\left({s_1, s_3}\right) \in \mathcal R_1$ and $\left({s_1, s_3}\right) \in \mathcal R_2$

and by definition of intersection:
 * $\left({s_1, s_3}\right) \in \mathcal R_1 \cap \mathcal R_2$

hence $\mathcal R_1 \cap \mathcal R_2$ is transitive.