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Syntax Definition cv 1618
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 2239. Since (when  y is distinct from  x) we have  x  =  { y  |  y  e.  x } by cvjust 2248, 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 1618 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1618 is intrinsically no different from any other class-building syntax such as cab 2239, cun 3076, or c0 3362.

For a general discussion of the theory of classes and the role of cv 1618, see http://us.metamath.org/mpegif/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 1620 of predicate calculus from the wceq 1619 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 1623.

Colors of variables: wff set class
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