Union of Antisymmetric Relation with Inverse is Antisymmetric iff Diagonal

Theorem
Let $\RR$ be an antisymmetric relation on a set $S$.

Then:
 * $\RR \cup \RR^{-1}$ is antisymmetric


 * $\RR = \Delta_S$

where:
 * $\RR^{-1}$ denotes the inverse of $\RR$
 * $\Delta_S$ denotes the diagonal relation
 * $\cup$ denotes set union.

Proof
As asserted, let $\RR$ be an antisymmetric relation.

Necessary Condition
Let $\RR = \Delta_S$.

From Inverse of Diagonal Relation,
 * $\RR^{-1} = \Delta_S$

Hence:
 * $\RR \cup \RR^{-1} = \Delta_S$

From Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation, we have that the diagonal relation $\Delta_S$ is antisymmetric.

Hence if $\RR = \Delta_S$, then $\RR \cup \RR^{-1}$ is antisymmetric.

Sufficient Condition
Let $\RR \cup \RR^{-1}$ be antisymmetric.

Let $\tuple {x, y} \in \RR$.

By definition of inverse relation:
 * $\tuple {x, y} \in \RR^{-1}$

Thus by definition of set union:
 * $\tuple {x, y} \in \RR \cup \RR^{-1}$

and
 * $\tuple {y, x} \in \RR \cup \RR^{-1}$

As $\RR \cup \RR^{-1}$ is antisymmetric:
 * $x = y$

Hence:
 * $\forall \tuple {x, y} \in \RR: x = y$

and it follows that:
 * $\RR = \Delta_S$