Intersection of Antisymmetric Relations is Antisymmetric

Theorem
The intersection of two antisymmetric relations is also an antisymmetric relation.

Proof
Let $\RR_1$ and $\RR_2$ be antisymmetric relations on a set $S$.

Let $\RR_3 = \RR_1 \cap \RR_2$.

Hence we have:

$\RR_3$ is not antisymmetric.

Then:

Hence we have:

That is, neither $\RR_1$ nor $\RR_2$ are antisymmetric.

From this contradiction it follows that $\RR_1 \cap \RR_2$ is an antisymmetric relation.