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Mirrors > Home > ILE Home > Th. List > diffitest | Unicode version |
Description: If subtracting any set from a finite set gives a finite set, any proposition of the form is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove . (Contributed by Jim Kingdon, 8-Sep-2021.) |
Ref | Expression |
---|---|
diffitest.1 |
Ref | Expression |
---|---|
diffitest |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4025 | . . . . . 6 | |
2 | snfig 6676 | . . . . . 6 | |
3 | 1, 2 | ax-mp 5 | . . . . 5 |
4 | diffitest.1 | . . . . 5 | |
5 | difeq1 3157 | . . . . . . . 8 | |
6 | 5 | eleq1d 2186 | . . . . . . 7 |
7 | 6 | albidv 1780 | . . . . . 6 |
8 | 7 | rspcv 2759 | . . . . 5 |
9 | 3, 4, 8 | mp2 16 | . . . 4 |
10 | rabexg 4041 | . . . . . 6 | |
11 | 3, 10 | ax-mp 5 | . . . . 5 |
12 | difeq2 3158 | . . . . . 6 | |
13 | 12 | eleq1d 2186 | . . . . 5 |
14 | 11, 13 | spcv 2753 | . . . 4 |
15 | 9, 14 | ax-mp 5 | . . 3 |
16 | isfi 6623 | . . 3 | |
17 | 15, 16 | mpbi 144 | . 2 |
18 | 0elnn 4502 | . . . . 5 | |
19 | breq2 3903 | . . . . . . . . . 10 | |
20 | en0 6657 | . . . . . . . . . 10 | |
21 | 19, 20 | syl6bb 195 | . . . . . . . . 9 |
22 | 21 | biimpac 296 | . . . . . . . 8 |
23 | rabeq0 3362 | . . . . . . . . 9 | |
24 | notrab 3323 | . . . . . . . . . 10 | |
25 | 24 | eqeq1i 2125 | . . . . . . . . 9 |
26 | 1 | snm 3613 | . . . . . . . . . 10 |
27 | r19.3rmv 3423 | . . . . . . . . . 10 | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 |
29 | 23, 25, 28 | 3bitr4i 211 | . . . . . . . 8 |
30 | 22, 29 | sylib 121 | . . . . . . 7 |
31 | 30 | olcd 708 | . . . . . 6 |
32 | ensym 6643 | . . . . . . . 8 | |
33 | elex2 2676 | . . . . . . . 8 | |
34 | enm 6682 | . . . . . . . 8 | |
35 | 32, 33, 34 | syl2an 287 | . . . . . . 7 |
36 | biidd 171 | . . . . . . . . . . . 12 | |
37 | 36 | elrab 2813 | . . . . . . . . . . 11 |
38 | 37 | simprbi 273 | . . . . . . . . . 10 |
39 | 38 | orcd 707 | . . . . . . . . 9 |
40 | 39, 24 | eleq2s 2212 | . . . . . . . 8 |
41 | 40 | exlimiv 1562 | . . . . . . 7 |
42 | 35, 41 | syl 14 | . . . . . 6 |
43 | 31, 42 | jaodan 771 | . . . . 5 |
44 | 18, 43 | sylan2 284 | . . . 4 |
45 | 44 | ancoms 266 | . . 3 |
46 | 45 | rexlimiva 2521 | . 2 |
47 | 17, 46 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 682 wal 1314 wceq 1316 wex 1453 wcel 1465 wral 2393 wrex 2394 crab 2397 cvv 2660 cdif 3038 c0 3333 csn 3497 class class class wbr 3899 com 4474 cen 6600 cfn 6602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1o 6281 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: (None) |
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