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

Definition df-sbc 3164
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 3189 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 3165 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 3165, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3164 in the form of sbc8g 3170. 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 3164 and assert that  [. A  /  x ]. ph is always false when  A is a proper class.

The theorem sbc2or 3171 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3165.

The related definition df-csb 3254 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 3163 . 2  wff  [. A  /  x ]. ph
51, 2cab 2424 . . 3  class  { x  |  ph }
63, 5wcel 1726 . 2  wff  A  e. 
{ x  |  ph }
74, 6wb 178 1  wff  ( [. A  /  x ]. ph  <->  A  e.  { x  |  ph }
)
Colors of variables: wff set class
This definition is referenced by:  dfsbcq  3165  dfsbcq2  3166  sbcex  3172  nfsbc1d  3180  nfsbcd  3183  cbvsbc  3191  sbcbid  3216  intab  4082  brab1  4260  iotacl  5444  riotasbc  6568  scottexs  7816  scott0s  7817  hta  7826  issubc  14040  dmdprd  15564  setinds  25410  bnj1454  29287  bnj110  29303
  Copyright terms: Public domain W3C validator