\begin{definition}[Definition:Class Membership]
To define membership not only for sets, but also for proper classes, we will extend the membership relation to include specific behaviors with proper classes and sets alike:
:$\forall A, B: \paren {A \in B \iff \exists x: \paren {A = x \land x \in B } }$
With this definition, no proper classes is a member of any other class, proper or not.
\end{definition}