![]() |
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 (see sbcex 3786). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3808 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 3778 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 3778, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3777 in the form of sbc8g 3784. 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 3777 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. Theorem sbc2or 3785 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3778. The related definition df-csb 3893 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 3776 | . 2 wff [𝐴 / 𝑥]𝜑 |
5 | 1, 2 | cab 2705 | . . 3 class {𝑥 ∣ 𝜑} |
6 | 3, 5 | wcel 2099 | . 2 wff 𝐴 ∈ {𝑥 ∣ 𝜑} |
7 | 4, 6 | wb 205 | 1 wff ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3778 dfsbcq2 3779 sbceqbid 3783 sbcex 3786 nfsbc1d 3794 nfsbcdw 3797 nfsbcd 3800 sbc5 3804 sbc6g 3806 cbvsbcw 3810 cbvsbcvw 3811 cbvsbc 3812 sbcieg 3817 sbcied 3822 sbcbid 3835 sbcbi2 3839 sbcimdv 3850 sbcg 3855 intab 4981 brab1 5196 iotacl 6534 riotasbc 7395 scottexs 9911 scott0s 9912 hta 9921 issubc 17821 dmdprd 19955 sbceqbidf 32298 bnj1454 34473 bnj110 34489 setinds 35374 bj-csbsnlem 36381 rdgssun 36857 frege54cor1c 43345 frege55lem1c 43346 frege55c 43348 |
Copyright terms: Public domain | W3C validator |