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Syntax Definition cv 1641
 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 {y ∣ y ∈ x} is a class by cab 2339. Since (when y is distinct from x) we have x = {y ∣ y ∈ x} by cvjust 2348, we can argue that the syntax "class x " can be viewed as an abbreviation for "class {y ∣ y ∈ 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 1641 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1641 is intrinsically no different from any other class-building syntax such as cab 2339, cun 3207, or c0 3550. For a general discussion of the theory of classes and the role of cv 1641, 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 1643 of predicate calculus from the wceq 1642 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 setvar x
Assertion
Ref Expression
cv class x