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

Syntax Definition cv 1394
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 2442. Since (when is distinct from ) we have by cvjust 2451, we can argue that the syntax "classx " can be viewed as an abbreviation for "class{ | 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 1394 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1394 is intrinsically no different from any other class-building syntax such as cab 2442, cun 3473, or c0 3784.

For a general discussion of the theory of classes and the role of cv 1394, see http://us.metamath.org/mpeuni/mmset.html#class.

(The description above applies to set theory, not predicate calculus. The purpose of introducing classx here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1733 of predicate calculus from the wceq 1395 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 No typesetting for: setvar x
Assertion
Ref Expression
cv No typesetting for: class x

See definition df-tru 1398 for more information.

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