Class is Proper iff Bijection from Class to Proper Class

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 between $A$ and $\mathrm P$.

Necessary Condition
Suppose that $A$ is a proper class.

Then by the Axiom of Limitation of Size there exists a bijection $f: A \to \mathrm U$, where $\mathrm U$ is the universal class.

There also exists a bijection $g: \mathrm P \to \mathrm U$.

By Inverse of Bijection is Bijection, there exists a bijection $g^{-1}: \mathrm U \to \mathrm P$

From Composite of Bijections is Bijection, it follows that $f \circ g^{-1}: A \to \mathrm P$ is a bijection.

Sufficient Condition
Suppose that there exists a bijection $f: A \to \mathrm P$.

Because $\mathrm P$ is a proper class, it follows from the Axiom of Limitation of Size that there exists a bijection $g: \mathrm P \to \mathrm U$.

Then by Composite of Bijections is Bijection, $f \circ g: A \to \mathrm U$ is a bijection from $A$ to the universe.

And so by the Axiom of Limitation of Size, $A$ is proper.

Hence the result.