Proper Class is not Element of Class

Theorem
Let $\mathrm P$ be a proper class.

Then $\mathrm P$ is not an element of any class, that is:
 * $\neg \exists A : \mathrm P \in A$

Proof
From the definition of a proper class, $\mathrm P$ is not a set.

The rest then follows from the definition of a class.