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Syntax Definition cv 1258
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 2042. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2051, 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 1258 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1258 is intrinsically no different from any other class-building syntax such as cab 2042, cun 2943, or c0 3252.

For a general discussion of the theory of classes and the role of cv 1258, see

(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 1408 of predicate calculus from the wceq 1259 of set theory, so that we don't overload the  = connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers.)

Ref Expression  setvar  x
Ref Expression
cv  class  x

See definition df-tru 1262 for more information.

Colors of variables: wff set class
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