MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-sbc Unicode version

Definition df-sbc 3328
Description: Define the proper substitution of a class for a set.

When is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3354 for our definition, which always evaluates to true for proper classes.

Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3329 below). For example, if is a proper class, Quine's substitution of for in evaluates to rather than our falsehood. (This can be seen by substituting , , and for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactical breakdown of , and it does not seem possible to express it with a single closed formula.

If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 3329, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3328 in the form of sbc8g 3335. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of in every use of this definition) we allow direct reference to df-sbc 3328 and assert that is always false when is a proper class.

The theorem sbc2or 3336 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3329.

The related definition df-csb 3435 defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)

Ref Expression

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3
2 vx . . 3
3 cA . . 3
41, 2, 3wsbc 3327 . 2
51, 2cab 2442 . . 3
63, 5wcel 1818 . 2
74, 6wb 184 1
Colors of variables: wff setvar class
This definition is referenced by:  dfsbcq  3329  dfsbcq2  3330  sbceqbid  3334  sbcex  3337  nfsbc1d  3345  nfsbcd  3348  cbvsbc  3356  sbcbi2  3378  sbcbid  3385  intab  4317  brab1  4497  iotacl  5579  riotasbc  6273  scottexs  8326  scott0s  8327  hta  8336  issubc  15204  dmdprd  17029  sbceqbidf  27380  setinds  29210  bnj1454  33900  bnj110  33916  bj-csbsnlem  34470  frege54cor1c  37942  frege55lem1c  37943  frege55c  37945
  Copyright terms: Public domain W3C validator