Inverse Relation Equal iff Subset

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


 * $\mathcal R \subseteq \mathcal R^{-1} \iff \mathcal R = \mathcal R^{-1} \iff \mathcal R^{-1} \subseteq \mathcal R$

Proof

 * Suppose $\mathcal R = \mathcal R^{-1}$.

Then from the definition of set equality, $\mathcal R \subseteq \mathcal R^{-1}$ and $\mathcal R^{-1} \subseteq \mathcal R$.


 * Suppose $\mathcal R \subseteq \mathcal R^{-1}$.

Then:

Thus $\mathcal R \subseteq \mathcal R^{-1} \implies \mathcal R^{-1} \subseteq \mathcal R$ and it follows that $\mathcal R = \mathcal R^{-1}$.


 * Suppose $\mathcal R^{-1} \subseteq \mathcal R$.

Then:

and the proof is complete.