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

Definition df-sbc 3122
Description: Define the proper substitution of a class for a set.

When  A is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3147 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 3123 below). For example, if  A is a proper class, Quine's substitution of 
A for  y in  0  e.  y evaluates to  0  e.  A rather than our falsehood. (This can be seen by substituting  A,  y, 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  ph, 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 3123, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3122 in the form of sbc8g 3128. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of  A in every use of this definition) we allow direct reference to df-sbc 3122 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3129 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3123.

The related definition df-csb 3212 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  |-  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)

Detailed syntax breakdown of Definition df-sbc
StepHypRef Expression
1 wph . . 3  wff  ph
2 vx . . 3  set  x
3 cA . . 3  class  A
41, 2, 3wsbc 3121 . 2  wff  [. A  /  x ]. ph
51, 2cab 2390 . . 3  class  { x  |  ph }
63, 5wcel 1721 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 177 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  3123  dfsbcq2  3124  sbcex  3130  nfsbc1d  3138  nfsbcd  3141  cbvsbc  3149  sbcbid  3174  intab  4040  brab1  4217  iotacl  5400  riotasbc  6524  scottexs  7767  scott0s  7768  hta  7777  issubc  13990  dmdprd  15514  setinds  25348  bnj1454  28919  bnj110  28935
  Copyright terms: Public domain W3C validator