Syntax Definition cv 1614

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 e. x} is a class by cab 2157. Since (when y is distinct from x) we have x = {y | y e. x} by cvjust 2165, we can argue that that the syntax "class x " can be viewed as an abbreviation for "class {y | y e. 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 1614 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1614 is intrinsically no different from any other class-building syntax such as cab 2157, cun 2857, or c0 3114.

(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 1616 of predicate calculus from the wceq 1615 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-5 1619.

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