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Mirrors > Home > MPE Home > Th. List > df-sbc | Structured version Visualization version Unicode version |
Description: Define the proper
substitution of a class for a set.
When is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3306 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 3281 below). For example, if is a proper class, Quine's substitution of for in evaluates to rather than our falsehood. (This can be seen by substituting , , and 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 , 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 3281, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3280 in the form of sbc8g 3287. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of in every use of this definition) we allow direct reference to df-sbc 3280 and assert that is always false when is a proper class. The theorem sbc2or 3288 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3281. The related definition df-csb 3376 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 | |
2 | vx | . . 3 | |
3 | cA | . . 3 | |
4 | 1, 2, 3 | wsbc 3279 | . 2 |
5 | 1, 2 | cab 2448 | . . 3 |
6 | 3, 5 | wcel 1898 | . 2 |
7 | 4, 6 | wb 189 | 1 |
Colors of variables: wff setvar class |
This definition is referenced by: dfsbcq 3281 dfsbcq2 3282 sbceqbid 3286 sbcex 3289 nfsbc1d 3297 nfsbcd 3300 cbvsbc 3308 sbcbi2 3328 sbcbid 3333 intab 4279 brab1 4462 iotacl 5588 riotasbc 6292 scottexs 8384 scott0s 8385 hta 8394 issubc 15789 dmdprd 17679 sbceqbidf 28166 bnj1454 29702 bnj110 29718 setinds 30473 bj-csbsnlem 31550 frege54cor1c 36556 frege55lem1c 36557 frege55c 36559 |
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