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Theorem abid1 2957
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.)

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

Proof of Theorem abid1
StepHypRef Expression
1 biid 264 . 2 (𝑥𝐴𝑥𝐴)
21abbi2i 2955 1 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  {cab 2802
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896
This theorem is referenced by:  abid2  2958  inrab2  4261  riotaclbgBAD  36195  aomclem4  39917
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