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Mirrors > Home > ILE Home > Th. List > ordsoexmid | Unicode version |
Description: Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.) |
Ref | Expression |
---|---|
ordsoexmid.1 |
Ref | Expression |
---|---|
ordsoexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtriexmidlem 4430 | . . . . 5 | |
2 | 1 | elexi 2693 | . . . 4 |
3 | 2 | sucid 4334 | . . 3 |
4 | 1 | onsuci 4427 | . . . 4 |
5 | suc0 4328 | . . . . 5 | |
6 | 0elon 4309 | . . . . . 6 | |
7 | 6 | onsuci 4427 | . . . . 5 |
8 | 5, 7 | eqeltrri 2211 | . . . 4 |
9 | eleq1 2200 | . . . . . . 7 | |
10 | 9 | 3anbi1d 1294 | . . . . . 6 |
11 | eleq1 2200 | . . . . . . 7 | |
12 | eleq1 2200 | . . . . . . . 8 | |
13 | 12 | orbi1d 780 | . . . . . . 7 |
14 | 11, 13 | imbi12d 233 | . . . . . 6 |
15 | 10, 14 | imbi12d 233 | . . . . 5 |
16 | 4 | elexi 2693 | . . . . . 6 |
17 | eleq1 2200 | . . . . . . . 8 | |
18 | 17 | 3anbi2d 1295 | . . . . . . 7 |
19 | eleq2 2201 | . . . . . . . 8 | |
20 | eleq2 2201 | . . . . . . . . 9 | |
21 | 20 | orbi2d 779 | . . . . . . . 8 |
22 | 19, 21 | imbi12d 233 | . . . . . . 7 |
23 | 18, 22 | imbi12d 233 | . . . . . 6 |
24 | p0ex 4107 | . . . . . . 7 | |
25 | eleq1 2200 | . . . . . . . . 9 | |
26 | 25 | 3anbi3d 1296 | . . . . . . . 8 |
27 | eleq2 2201 | . . . . . . . . . 10 | |
28 | eleq1 2200 | . . . . . . . . . 10 | |
29 | 27, 28 | orbi12d 782 | . . . . . . . . 9 |
30 | 29 | imbi2d 229 | . . . . . . . 8 |
31 | 26, 30 | imbi12d 233 | . . . . . . 7 |
32 | ordsoexmid.1 | . . . . . . . . . . 11 | |
33 | df-iso 4214 | . . . . . . . . . . 11 | |
34 | 32, 33 | mpbi 144 | . . . . . . . . . 10 |
35 | 34 | simpri 112 | . . . . . . . . 9 |
36 | epel 4209 | . . . . . . . . . . . 12 | |
37 | epel 4209 | . . . . . . . . . . . . 13 | |
38 | epel 4209 | . . . . . . . . . . . . 13 | |
39 | 37, 38 | orbi12i 753 | . . . . . . . . . . . 12 |
40 | 36, 39 | imbi12i 238 | . . . . . . . . . . 11 |
41 | 40 | 2ralbii 2441 | . . . . . . . . . 10 |
42 | 41 | ralbii 2439 | . . . . . . . . 9 |
43 | 35, 42 | mpbi 144 | . . . . . . . 8 |
44 | 43 | rspec3 2520 | . . . . . . 7 |
45 | 24, 31, 44 | vtocl 2735 | . . . . . 6 |
46 | 16, 23, 45 | vtocl 2735 | . . . . 5 |
47 | 2, 15, 46 | vtocl 2735 | . . . 4 |
48 | 1, 4, 8, 47 | mp3an 1315 | . . 3 |
49 | 2 | elsn 3538 | . . . . 5 |
50 | ordtriexmidlem2 4431 | . . . . 5 | |
51 | 49, 50 | sylbi 120 | . . . 4 |
52 | elirr 4451 | . . . . . . 7 | |
53 | elrabi 2832 | . . . . . . 7 | |
54 | 52, 53 | mto 651 | . . . . . 6 |
55 | elsuci 4320 | . . . . . . 7 | |
56 | 55 | ord 713 | . . . . . 6 |
57 | 54, 56 | mpi 15 | . . . . 5 |
58 | 0ex 4050 | . . . . . . 7 | |
59 | biidd 171 | . . . . . . 7 | |
60 | 58, 59 | rabsnt 3593 | . . . . . 6 |
61 | 60 | eqcoms 2140 | . . . . 5 |
62 | 57, 61 | syl 14 | . . . 4 |
63 | 51, 62 | orim12i 748 | . . 3 |
64 | 3, 48, 63 | mp2b 8 | . 2 |
65 | orcom 717 | . 2 | |
66 | 64, 65 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 w3a 962 wceq 1331 wcel 1480 wral 2414 crab 2418 c0 3358 csn 3522 class class class wbr 3924 cep 4204 wpo 4211 wor 4212 con0 4280 csuc 4282 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-tr 4022 df-eprel 4206 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 |
This theorem is referenced by: (None) |
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