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 Reflexive contains Diagonal Relation, we have that:
 * $$\Delta_S \subseteq \mathcal R_1$$;
 * $$\Delta_S \subseteq \mathcal R_2$$.

Hence from Intersection Largest, $$\Delta_S \subseteq \mathcal R_1 \cap \mathcal R_2$$.

Hence the result, from Reflexive contains Diagonal Relation.