| 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 2724. The proof does not rely on cvjust 2724, so cvjust 2724
could be proved as a special instance of it. Note however that abid1 2865
necessarily relies on df-clel 2804, whereas cvjust 2724 does not.
This theorem requires ax-ext 2702, df-clab 2709, df-cleq 2722, df-clel 2804, 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 1539, cab 2708,
and statements corresponding to defined class constructors.
Note on the simultaneous presence in set.mm of this abid1 2865 and its
commuted form abid2 2866: 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 2753 (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 (left-hand side), and a "computational
form",
where the shorter expression is on the RHS (right-hand side). An
example is df-2 12256 versus 1p1e2 12313. We do not need 1p1e2 12313, 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 12331, 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 pm4.24 563 and anidm 564, 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 2865 and abid2 2866 are in set.mm.
(Contributed by NM, 26-Dec-1993.) (Revised by BJ,
10-Nov-2020.) |
