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 2819. The proof does not rely on cvjust 2819, so cvjust 2819
could be proved as a special instance of it. Note however that abid1 2957
necessarily relies on df-clel 2896, whereas cvjust 2819 does not.
This theorem requires ax-ext 2796, df-clab 2803, df-cleq 2817, df-clel 2896, 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 1537, cab 2802,
and statements corresponding to defined class constructors.
Note on the simultaneous presence in set.mm of this abid1 2957 and its
commuted form abid2 2958: 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 2847 (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 11697 versus 1p1e2 11759. We do not need 1p1e2 11759, 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 11777, 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 pm4.24 567 and anidm 568, 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 2957 and abid2 2958 are in set.mm.
(Contributed by NM, 26-Dec-1993.) (Revised by BJ,
10-Nov-2020.) |