| Description: Every class is equal to a
class abstraction (the class of sets belonging
to it). Theorem 5.2 of [Quine] p. 35.
This is a generalization to
classes of cvjust 2816. The proof does not rely on cvjust 2816, so cvjust 2816
could be proved as a special instance of it. Note however that abid1 2956
necessarily relies on df-clel 2893, whereas cvjust 2816 does not.
This theorem requires ax-ext 2793, df-clab 2800, df-cleq 2814, df-clel 2893, but
to prove that any specific class term not containing class variables is
a setvar or is equal to a class abstraction does not require these
$a-statements. This last fact is a metatheorem, consequence of the fact
that the only $a-statements with typecode class are cv 1536, cab 2799,
and statements corresponding to defined class constructors.
Note on the simultaneous presence in set.mm of this abid1 2956 and its
commuted form abid2 2957: It is rare that two forms so closely
related
both appear in set.mm. Indeed, such equalities are generally used in
later proofs as parts of transitive inferences, and with the many
variants of eqtri 2844 (search for *eqtr*), it would be rare that
either
one would shorten a proof compared to the other. There is typically a
choice between what we call a "definitional form", where the
shorter
expression is on the LHS, and a "computational form", where
the shorter
expression is on the RHS. An example is df-2 11701
versus 1p1e2 11763. We do
not need 1p1e2 11763, but because it occurs "naturally"
in computations, it
can be useful to have it directly, together with a uniform set of
1-digit operations like 1p2e3 11781, etc. In most cases, we do not need
both a definitional and a computational forms. A definitional form
would favor consistency with genuine definitions, while a computational
form is often more natural. The situation is similar with
biconditionals in propositional calculus: see for instance pm4.24 566 and
anidm 567, while other biconditionals generally appear
in a single form
(either definitional, but more often computational). In the present
case, the equality is important enough that both abid1 2956 and abid2 2957 are
in set.mm.
(Contributed by NM, 26-Dec-1993.) (Revised by BJ,
10-Nov-2020.) |
