Intersection of Orderings is Ordering

Theorem
Let $A$ be a set.

Let $\RR$ and $\SS$ be orderings on $A$.

Then $\RR \cap \SS$ is an ordering on $A$.

Proof
By definition of ordering:


 * $\RR$ and $\SS$ are reflexive


 * $\RR$ and $\SS$ are transitive


 * $\RR$ and $\SS$ are antisymmetric.

We have:
 * Intersection of Reflexive Relations is Reflexive
 * Intersection of Transitive Relations is Transitive
 * Intersection of Antisymmetric Relations is Antisymmetric

and the result follows.