MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abid1 Structured version   Visualization version   GIF version

Theorem abid1 2730
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 2604. The proof does not rely on cvjust 2604, so cvjust 2604 could be proved as a special instance of it. Note however that abid1 2730 necessarily relies on df-clel 2605, whereas cvjust 2604 does not.

This theorem requires ax-ext 2589, df-clab 2596, df-cleq 2602, df-clel 2605, 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 1473, cab 2595 and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2730 and its commuted form abid2 2731: 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 2631 (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 10928 versus 1p1e2 10983. We do not need 1p1e2 10983, 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 11001, 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 pm4.24 672 and anidm 673, 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 2730 and abid2 2731 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 249 . 2 (𝑥𝐴𝑥𝐴)
21abbi2i 2724 1 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  {cab 2595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605
This theorem is referenced by:  abid2  2731  inrab2  3858  riotaclbgBAD  33041  aomclem4  36428
  Copyright terms: Public domain W3C validator