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 if and only if there exists a bijection from $P$ to $A$.

Proof

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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 $P$ if and only if there exists a bijection from $P$ to $A$.

The rest follows from Inverse of Bijection is Bijection.

$\blacksquare$