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