Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-sbc | Structured version Visualization version GIF version |
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 3449 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 3424 below). For example, if 𝐴 is a proper class, Quine's substitution of 𝐴 for 𝑦 in 0 ∈ 𝑦 evaluates to 0 ∈ 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 syntactic breakdown of 𝜑, 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 3424, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3423 in the form of sbc8g 3430. 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 3423 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3431 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3424. The related definition df-csb 3520 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 |
---|---|
df-sbc | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cA | . . 3 class 𝐴 | |
4 | 1, 2, 3 | wsbc 3422 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2607 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 1987 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 196 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3424 dfsbcq2 3425 sbceqbid 3429 sbcex 3432 nfsbc1d 3440 nfsbcd 3443 cbvsbc 3451 sbcbi2 3471 sbcbid 3476 intab 4479 brab1 4670 iotacl 5843 riotasbc 6591 scottexs 8710 scott0s 8711 hta 8720 issubc 16435 dmdprd 18337 sbceqbidf 29210 bnj1454 30673 bnj110 30689 setinds 31437 bj-csbsnlem 32598 frege54cor1c 37730 frege55lem1c 37731 frege55c 37733 |
Copyright terms: Public domain | W3C validator |