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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation: Theorem $1.1$