Not Every Class is a Set/Proof 2

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Theorem

Let $A$ be a class.

Then it is not necessarily the case that $A$ is also a set.


Proof

Consider the universal class $V$.

From Class has Subclass which is not Element, $V$ has a subclass $A$ which is not an element of $V$.

But $V$ contains as elements all the sets.

It follows that $A$ is a class which is not a set.

$\blacksquare$


Sources