Reflexive Reduction of Transitive Relation is Transitive

Theorem
Let $S$ be a set.

Let $\RR$ be a transitive relation on $S$.

Let $\RR^\ne$ be the reflexive reduction of $\RR$.

Then $\RR^\ne$ is transitive.

Proof
Let $a, b, c \in S$.

Let $a \mathrel {\RR^\ne} b$ and $b \mathrel {\RR^\ne} c$.

By the definition of reflexive reduction:


 * $a \ne b$ and $a \mathrel \RR b$
 * $b \ne c$ and $b \mathrel \RR c$

Since $\mathrel \RR$ is transitive:


 * $a \mathrel \RR c$