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Syntax Definition cv 1397
Description: This syntax construction states that a variable x, which has been declared to be a set 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 1882. Since (when y is distinct from x) we have x = {y y x} by cvjust 1890, we can argue that 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 1397 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1397 is intrinsically no different from any other class-building syntax such as cab 1882, cun 2609, or c0 2726.

(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 1399 of predicate calculus from the wceq 1398 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 set x
Assertion
Ref Expression
cv class x

This syntax is primitive. The first axiom using it is ax-8 1402.

Colors of variables: wff set class
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