Intersection of Reflexive Relations is Reflexive

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

Proof
Let $\mathcal R_1$ and $\mathcal R_2$ be reflexive relations on a set $S$.

From Relation Contains Diagonal Relation iff Reflexive, we have that:
 * $\Delta_S \subseteq \mathcal R_1$
 * $\Delta_S \subseteq \mathcal R_2$

Hence from Intersection is Largest Subset:
 * $\Delta_S \subseteq \mathcal R_1 \cap \mathcal R_2$

Hence the result, from Relation Contains Diagonal Relation iff Reflexive.