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

Theorem abid1 2879
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 2732. The proof does not rely on cvjust 2732, so cvjust 2732 could be proved as a special instance of it. Note however that abid1 2879 necessarily relies on df-clel 2817, whereas cvjust 2732 does not.

This theorem requires ax-ext 2709, df-clab 2716, df-cleq 2730, df-clel 2817, 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 1542, cab 2715, and statements corresponding to defined class constructors.

Note on the simultaneous presence in set.mm of this abid1 2879 and its commuted form abid2 2880: 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 2766 (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 11918 versus 1p1e2 11980. We do not need 1p1e2 11980, 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 11998, 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 2879 and abid2 2880 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 2877 1 𝐴 = {𝑥𝑥𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2111  {cab 2715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817
This theorem is referenced by:  abid2  2880  inrab2  4237  nsgqusf1olem2  31340  riotaclbgBAD  36735  aomclem4  40618
  Copyright terms: Public domain W3C validator