Metamath Proof Explorer

Syntax definition wcel

Description: Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see mmset.html#class .

(The purpose of introducing wff A e. B here is to allow us to express, i.e., "prove", the wel of predicate calculus in terms of the wcel of set theory, so that we do not "overload" the e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab for more information on the set theory usage of wcel .)

Ref Expression
Assertion wcel wff A B