# 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.

## Also see

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

## Sources

- 1963: Willard Van Orman Quine:
*Set Theory and Its Logic*: $\S 6.3$