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

Theorem
Let $A$ be a class.

Let $P$ be a proper class.

Then $A$ is proper there exists a bijection from $P$ 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 $P$ there exists a bijection from $P$ to $A$.

The rest follows from Inverse of Bijection is Bijection.