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Theorem abid1 2866
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 2722. The proof does not rely on cvjust 2722, so cvjust 2722 could be proved as a special instance of it. Note however that abid1 2866 necessarily relies on df-clel 2806, whereas cvjust 2722 does not.

This theorem requires ax-ext 2699, df-clab 2706, df-cleq 2720, df-clel 2806, 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 1533, cab 2705, and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2866 and its commuted form abid2 2867: 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 2756 (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 12306 versus 1p1e2 12368. We do not need 1p1e2 12368, 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 12386, 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 2866 and abid2 2867 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 2865 1 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {cab 2705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806
This theorem is referenced by:  abid2  2867  eqab  2868  eqabb  2869  abssdv  4063  inrab2  4308  nsgqusf1olem2  33137  disjdmqscossss  38275  riotaclbgBAD  38426  ssabdv  41708  aomclem4  42481  limexissupab  42712
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