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Syntax Definition cv 1222
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 {yy x} is a class by cab 1999. Since (when y is distinct from x) we have x = {yy x} by cvjust 2008, we can argue that the syntax "class x " can be viewed as an abbreviation for "class {yy 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 1222 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1222 is intrinsically no different from any other class-building syntax such as cab 1999, cun 2883, or c0 3192.

For a general discussion of the theory of classes and the role of cv 1222, 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 1365 of predicate calculus from the wceq 1223 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 1226 for more information.

Colors of variables: wff set class
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