Proper Class is not Element of Class

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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.

$\blacksquare$