User:Dfeuer/Universal Class is not Set
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Theorem
Let $\mathbb U$ be the universal class.
Then $\mathbb U$ is not a set.
That is, $\mathbb U \notin \mathbb U$.
Proof
By User:Dfeuer/Not Every Class is a Set, there is a class $A$ which is not a set.
That is, $A \subseteq \mathbb U$ but $A \notin \mathbb U$.
Suppose for the sake of contradiction that $\mathbb U$ is a set.
That is, suppose that $\mathbb U \in \mathbb U$.
By User:Dfeuer/Subclass of Set is Set, $A$ is a set. That is, $A \in \mathbb U$, a contradiction.
$\blacksquare$