Characterization of 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.

Also see

 * Universal Class
 * Class in ZF, where class membership is taken to be a definitional abbreviation

Source

 * : $\S 6.3$