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

Syntax Definition cv 1467
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 {𝑦𝑦𝑥} is a class by cab 2491. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2500, we can argue that the syntax "class 𝑥 " can be viewed as an abbreviation for "class {𝑦𝑦𝑥}". 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 1467 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1467 is intrinsically no different from any other class-building syntax such as cab 2491, cun 3424, or c0 3757.

For a general discussion of the theory of classes and the role of cv 1467, 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 1822 of predicate calculus from the wceq 1468 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 𝑥
Assertion
Ref Expression
cv class 𝑥

See definition df-tru 1471 for more information.

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