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Theorem abid1 2862
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 2718. The proof does not rely on cvjust 2718, so cvjust 2718 could be proved as a special instance of it. Note however that abid1 2862 necessarily relies on df-clel 2802, whereas cvjust 2718 does not.

This theorem requires ax-ext 2695, df-clab 2702, df-cleq 2716, df-clel 2802, 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 1532, cab 2701, and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2862 and its commuted form abid2 2863: 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 2752 (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 12272 versus 1p1e2 12334. We do not need 1p1e2 12334, 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 12352, 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 2862 and abid2 2863 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 261 . 2 (𝑥𝐴𝑥𝐴)
21eqabi 2861 1 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802
This theorem is referenced by:  abid2  2863  eqab  2864  eqabb  2865  abssdv  4057  inrab2  4299  nsgqusf1olem2  32994  disjdmqscossss  38163  riotaclbgBAD  38314  ssabdv  41530  aomclem4  42288  limexissupab  42522
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