Definition:Class Membership

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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:

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

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

Justification

With this definition, no proper classes is a member of any class, since they are not equal to another set.

This definition only establishes a particular behavior for proper classes.