MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cv Unicode version

Syntax Definition cv 1622
Description: This syntax construction states that a variable  x, which has been declared to be a set 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 2269. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2278, we can argue that 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 1622 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1622 is intrinsically no different from any other class-building syntax such as cab 2269, cun 3150, or c0 3455.

For a general discussion of the theory of classes and the role of cv 1622, 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 1624 of predicate calculus from the wceq 1623 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
vx  set  x
Ref Expression
cv  class  x

See definition df-sb 1630 for more information.

Colors of variables: wff set class
  Copyright terms: Public domain W3C validator