Metamath Proof Explorer

Syntax definition cv

Description: This syntax construction states that a variable x , which has been declared to be a setvar variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder { y | y e. x } is a class by cab . Since (when y is distinct from x ) we have x = { y | y e. x } by cvjust , we can argue that the syntax " class x " can be viewed as an abbreviation for " class { y | y e. x } ". See the discussion under the definition of class in Jech p. 4 showing that "Every set can be considered to be a class".

While it is tempting and perhaps occasionally useful to view cv as a "type conversion" from a setvar variable to a class variable, keep in mind that cv is intrinsically no different from any other class-building syntax such as cab , cun , or c0 .

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

(The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express, i.e., "prove", the weq of predicate calculus from the wceq of set theory, so that we do not overload the = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

Ref Expression
Assertion cv setvar x