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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 4106 | . . . . . 6 | |
2 | snfig 6774 | . . . . . 6 | |
3 | 1, 2 | ax-mp 5 | . . . . 5 |
4 | diffitest.1 | . . . . 5 | |
5 | difeq1 3231 | . . . . . . . 8 | |
6 | 5 | eleq1d 2233 | . . . . . . 7 |
7 | 6 | albidv 1811 | . . . . . 6 |
8 | 7 | rspcv 2824 | . . . . 5 |
9 | 3, 4, 8 | mp2 16 | . . . 4 |
10 | rabexg 4122 | . . . . . 6 | |
11 | 3, 10 | ax-mp 5 | . . . . 5 |
12 | difeq2 3232 | . . . . . 6 | |
13 | 12 | eleq1d 2233 | . . . . 5 |
14 | 11, 13 | spcv 2818 | . . . 4 |
15 | 9, 14 | ax-mp 5 | . . 3 |
16 | isfi 6721 | . . 3 | |
17 | 15, 16 | mpbi 144 | . 2 |
18 | 0elnn 4593 | . . . . 5 | |
19 | breq2 3983 | . . . . . . . . . 10 | |
20 | en0 6755 | . . . . . . . . . 10 | |
21 | 19, 20 | bitrdi 195 | . . . . . . . . 9 |
22 | 21 | biimpac 296 | . . . . . . . 8 |
23 | rabeq0 3436 | . . . . . . . . 9 | |
24 | notrab 3397 | . . . . . . . . . 10 | |
25 | 24 | eqeq1i 2172 | . . . . . . . . 9 |
26 | 1 | snm 3693 | . . . . . . . . . 10 |
27 | r19.3rmv 3497 | . . . . . . . . . 10 | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . 9 |
29 | 23, 25, 28 | 3bitr4i 211 | . . . . . . . 8 |
30 | 22, 29 | sylib 121 | . . . . . . 7 |
31 | 30 | olcd 724 | . . . . . 6 |
32 | ensym 6741 | . . . . . . . 8 | |
33 | elex2 2740 | . . . . . . . 8 | |
34 | enm 6780 | . . . . . . . 8 | |
35 | 32, 33, 34 | syl2an 287 | . . . . . . 7 |
36 | biidd 171 | . . . . . . . . . . . 12 | |
37 | 36 | elrab 2880 | . . . . . . . . . . 11 |
38 | 37 | simprbi 273 | . . . . . . . . . 10 |
39 | 38 | orcd 723 | . . . . . . . . 9 |
40 | 39, 24 | eleq2s 2259 | . . . . . . . 8 |
41 | 40 | exlimiv 1585 | . . . . . . 7 |
42 | 35, 41 | syl 14 | . . . . . 6 |
43 | 31, 42 | jaodan 787 | . . . . 5 |
44 | 18, 43 | sylan2 284 | . . . 4 |
45 | 44 | ancoms 266 | . . 3 |
46 | 45 | rexlimiva 2576 | . 2 |
47 | 17, 46 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 698 wal 1340 wceq 1342 wex 1479 wcel 2135 wral 2442 wrex 2443 crab 2446 cvv 2724 cdif 3111 c0 3407 csn 3573 class class class wbr 3979 com 4564 cen 6698 cfn 6700 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-iinf 4562 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-br 3980 df-opab 4041 df-id 4268 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-1o 6378 df-er 6495 df-en 6701 df-fin 6703 |
This theorem is referenced by: (None) |
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