# Definition:Class Membership

## Definition

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:

$\forall A,B: ( A \in B \iff \exists x: ( A = x \land x \in B ) )$

With this definition, no proper class is a member of any other class, proper or not.

## Justification

With this definition, no proper class is a member of any class, since they are not equal to another set. This definition only establishes a particular behavior for proper classes.