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