ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cv Structured version   Unicode version

Syntax Definition cv 1241
Description: This syntax construction states that a variable , 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  {  |  } is a class by cab 2023. Since (when is distinct from ) we have  {  |  } by cvjust 2032, we can argue that the syntax " " can be viewed as an abbreviation for "  {  |  }". 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 1241 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1241 is intrinsically no different from any other class-building syntax such as cab 2023, cun 2909, or c0 3218.

For a general discussion of the theory of classes and the role of cv 1241, see http://us.metamath.org/mpeuni/mmset.html#class.

(The description above applies to set theory, not predicate calculus. The purpose of introducing here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1389 of predicate calculus from the wceq 1242 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.)

Hypothesis
Ref Expression
vx.cv  setvar
Assertion
Ref Expression
cv

See definition df-tru 1245 for more information.

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