Intersection of Transitive Relations is Transitive/General Result

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Theorem

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


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


Proof