User:Dfeuer/Not Every Class is a Set

Theorem

Not every class is a set.

That is, if $\mathbb U$ is the universal class, then:

$\exists B: B \subseteq \mathbb U \land B \notin \mathbb U$

Proof

Define a first-order formula $\varphi$ by letting

$\varphi(x) \iff \lnot(x \in x)$

Then by the Axiom Schema of Separation, there is a class $O \subseteq \mathbb U$ such that

$\forall x: x \in O \iff \varphi(x)$

That is:

$\forall x: x \in O \iff x \notin x$

Suppose for the sake of contradiction that $O$ is a set.

That is, suppose that $O \in \mathbb U$.

Then $O \in O \iff O \notin O$, a contradiction.

Thus $O$ is not a set.

$\blacksquare$