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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 (see sbcex 3782). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3802 for our definition, whose right-hand side 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 3774 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 3774, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3773 in the form of sbc8g 3780. 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 3773 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The theorem sbc2or 3781 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3774. The related definition df-csb 3884 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 3772 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2799 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2114 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 208 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3774 dfsbcq2 3775 sbceqbid 3779 sbcex 3782 nfsbc1d 3790 nfsbcdw 3793 nfsbcd 3796 cbvsbcw 3804 cbvsbc 3806 sbcbid 3826 sbcbi2OLD 3832 intab 4906 brab1 5114 iotacl 6341 riotasbc 7132 scottexs 9316 scott0s 9317 hta 9326 issubc 17105 dmdprd 19120 sbceqbidf 30250 bnj1454 32114 bnj110 32130 setinds 33023 bj-csbsnlem 34223 rdgssun 34662 frege54cor1c 40281 frege55lem1c 40282 frege55c 40284 |
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