Inverse Relation Equal iff Subset

Theorem
If a relation $\mathcal R$ is a subset or superset of its inverse, then it equals its inverse.

That is, the following are equivalent:

$(1)$ implies $(2)$
Suppose $\mathcal R \subseteq \mathcal R^{-1}$.

Then:

Hence $\mathcal R^{-1} \subseteq \mathcal R$.

$(2)$ implies $(1)$
Since $(1)$ implies $(2)$ was already established above, interpreting it on $\mathcal R^{-1}$ yields:


 * $\mathcal R^{-1} \subseteq \left({\mathcal R^{-1}}\right)^{-1}$

By Inverse of Inverse Relation, $\left({\mathcal R^{-1}}\right)^{-1} = \mathcal R$.

Hence $(2)$ implies $(1)$.

$(3)$ iff $(1)$ and $(2)$
This is precisely the definition of set equality.