**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.) |