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Syntax Definition cv 1286
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 2071. Since (when 𝑦 is distinct from 𝑥) we have 𝑥 = {𝑦𝑦𝑥} by cvjust 2080, 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 1286 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1286 is intrinsically no different from any other class-building syntax such as cab 2071, cun 2986, or c0 3275.

For a general discussion of the theory of classes and the role of cv 1286, see http://us.metamath.org/mpeuni/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 1435 of predicate calculus from the wceq 1287 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 1290 for more information.

Colors of variables: wff set class
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