Characterization of Class Membership

Theorem

Let $A$ and $B$ be classes.

Then:

$\forall A, B: \paren {A \in B \iff \exists x: \paren {A = x \land x \in B} }$

where $x$ is specifically a set.

Let $V$ denote the universal class.

By Class is Subclass of Universal Class, $A \subseteq V$ and $B \subseteq V$.

By definition of universal class, every element of $V$ is a set.

Hence every element of $B$ is a set.

So if $A \in B$, then it follows that $A$ is itself a set.

Hence the result.

$\blacksquare$

Definition:Universal Class
Definition:Class (Zermelo-Fraenkel), where class membership is taken to be a definitional abbreviation