Class is Proper iff Bijection from Class to Proper Class/Corollary

Theorem
Let $A$ be a class.

Let $\mathrm P$ be a proper class.

Then $A$ is proper if and only if there exists a bijection from $\mathrm P$ to $A$.

Proof
From Biconditional is Transitive and Class is Proper iff Bijection from Class to Proper Class, it suffices to show that:
 * There exists a bijection from $A$ to $\mathrm P$ iff there exists a bijection from $\mathrm P$ to $A$.

The rest follows from Inverse of Bijection is Bijection.