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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 3429 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 3404 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 3404, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3403 in the form of sbc8g 3410. 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 3403 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3411 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3404. The related definition df-csb 3500 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 3402 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2596 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 1977 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 195 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3404 dfsbcq2 3405 sbceqbid 3409 sbcex 3412 nfsbc1d 3420 nfsbcd 3423 cbvsbc 3431 sbcbi2 3451 sbcbid 3456 intab 4442 brab1 4630 iotacl 5791 riotasbc 6526 scottexs 8633 scott0s 8634 hta 8643 issubc 16318 dmdprd 18220 sbceqbidf 28705 bnj1454 30166 bnj110 30182 setinds 30927 bj-csbsnlem 32090 frege54cor1c 37229 frege55lem1c 37230 frege55c 37232 |
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