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

Definition df-sbc 3171
 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 3196 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 3172 below). For example, if is a proper ?Error: 0 e. y ^ This math symbol is not active (i.e. was not declared in this scope). class, Quine's substitution of for in 0e. evaluates to ?Error: 0 e. A ^ This math symbol is not active (i.e. was not declared in this scope). 0e. rather than our falsehood. (This can be seen by substituting ?Error: 0 ^ This math symbol is not active (i.e. was not declared in this scope). , , and 0 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 3172, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3171 in the form of sbc8g 3177. 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 3171 and assert that is always false when is a proper class. The theorem sbc2or 3178 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3172. The related definition df-csb 3271 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.)
Assertion
Ref Expression
df-sbc

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3
2 vx . . 3
3 cA . . 3
41, 2, 3wsbc 3170 . 2
51, 2cab 2429 . . 3
63, 5wcel 1728 . 2
74, 6wb 178 1
 Colors of variables: wff set class This definition is referenced by:  dfsbcq  3172  dfsbcq2  3173  sbcex  3179  nfsbc1d  3187  nfsbcd  3190  cbvsbc  3198  sbcbid  3226  intab  4109  brab1  4288  iotacl  5487  riotasbc  6615  scottexs  7862  scott0s  7863  hta  7872  issubc  14086  dmdprd  15610  setinds  25509  sbcbi2  26849  bnj1454  29454  bnj110  29470
 Copyright terms: Public domain W3C validator