Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest/Proof 1

Proof
Let $\RR^+$ be the intersection of all transitive relations containing $\RR$.

By Trivial Relation is Equivalence:
 * the trivial relation $S \times S$ is transitive.

By the definition of endorelation:
 * $\RR \subseteq S \times S$.

Next we have that the Intersection of Transitive Relations is Transitive.

Thus $\RR^+$ is the smallest transitive relation on $S$ containing $\RR$.