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Theorem abid1 2928
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 2801. The proof does not rely on cvjust 2801, so cvjust 2801 could be proved as a special instance of it. Note however that abid1 2928 necessarily relies on df-clel 2802, whereas cvjust 2801 does not.

This theorem requires ax-ext 2784, df-clab 2793, df-cleq 2799, df-clel 2802, but to prove that any specific class term not containing class variables is a setvar or can be written as (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 1636, cab 2792 and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2928 and its commuted form abid2 2929: 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 2828 (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, and a "computational form" where the shorter expression is on the rhs. An example is df-2 11364 versus 1p1e2 11417. We do not need 1p1e2 11417, 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 11435, 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 computationa form is often more natural. The situation is similar with biconditionals in propositional calculus: see for instance pm4.24 555 and anidm 556, 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 2928 and abid2 2929 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 252 . 2 (𝑥𝐴𝑥𝐴)
21abbi2i 2922 1 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wcel 2156  {cab 2792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802
This theorem is referenced by:  abid2  2929  inrab2  4101  riotaclbgBAD  34733  aomclem4  38128
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