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Syntax Definition cv 1334
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 2143. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2152, 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 1334 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1334 is intrinsically no different from any other class-building syntax such as cab 2143, cun 3100, or c0 3394.

For a general discussion of the theory of classes and the role of cv 1334, see https://us.metamath.org/mpeuni/mmset.html#class 1334.

(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 1483 of predicate calculus from the wceq 1335 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 1338 for more information.

Colors of variables: wff set class
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