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Theorem abid1 2870
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 2726. The proof does not rely on cvjust 2726, so cvjust 2726 could be proved as a special instance of it. Note however that abid1 2870 necessarily relies on df-clel 2810, whereas cvjust 2726 does not.

This theorem requires ax-ext 2703, df-clab 2710, df-cleq 2724, df-clel 2810, 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 1540, cab 2709, and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2870 and its commuted form abid2 2871: 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 2760 (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 12274 versus 1p1e2 12336. We do not need 1p1e2 12336, 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 12354, 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 pm4.24 564 and anidm 565, 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 2870 and abid2 2871 are in set.mm.

(Contributed by NM, 26-Dec-1993.) (Revised by BJ, 10-Nov-2020.)

Assertion
Ref Expression
abid1 𝐴 = {𝑥𝑥𝐴}
Distinct variable group:   𝑥,𝐴

Proof of Theorem abid1
StepHypRef Expression
1 biid 260 . 2 (𝑥𝐴𝑥𝐴)
21eqabi 2869 1 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  {cab 2709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810
This theorem is referenced by:  abid2  2871  eqab  2872  eqabb  2873  abssdv  4065  inrab2  4307  nsgqusf1olem2  32520  disjdmqscossss  37668  riotaclbgBAD  37819  ssabdv  41039  aomclem4  41789  limexissupab  42023
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