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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 4435 | . . . . 5 | |
2 | 1 | elexi 2698 | . . . 4 |
3 | 2 | sucid 4339 | . . 3 |
4 | 1 | onsuci 4432 | . . . 4 |
5 | suc0 4333 | . . . . 5 | |
6 | 0elon 4314 | . . . . . 6 | |
7 | 6 | onsuci 4432 | . . . . 5 |
8 | 5, 7 | eqeltrri 2213 | . . . 4 |
9 | eleq1 2202 | . . . . . . 7 | |
10 | 9 | 3anbi1d 1294 | . . . . . 6 |
11 | eleq1 2202 | . . . . . . 7 | |
12 | eleq1 2202 | . . . . . . . 8 | |
13 | 12 | orbi1d 780 | . . . . . . 7 |
14 | 11, 13 | imbi12d 233 | . . . . . 6 |
15 | 10, 14 | imbi12d 233 | . . . . 5 |
16 | 4 | elexi 2698 | . . . . . 6 |
17 | eleq1 2202 | . . . . . . . 8 | |
18 | 17 | 3anbi2d 1295 | . . . . . . 7 |
19 | eleq2 2203 | . . . . . . . 8 | |
20 | eleq2 2203 | . . . . . . . . 9 | |
21 | 20 | orbi2d 779 | . . . . . . . 8 |
22 | 19, 21 | imbi12d 233 | . . . . . . 7 |
23 | 18, 22 | imbi12d 233 | . . . . . 6 |
24 | p0ex 4112 | . . . . . . 7 | |
25 | eleq1 2202 | . . . . . . . . 9 | |
26 | 25 | 3anbi3d 1296 | . . . . . . . 8 |
27 | eleq2 2203 | . . . . . . . . . 10 | |
28 | eleq1 2202 | . . . . . . . . . 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 4219 | . . . . . . . . . . 11 | |
34 | 32, 33 | mpbi 144 | . . . . . . . . . 10 |
35 | 34 | simpri 112 | . . . . . . . . 9 |
36 | epel 4214 | . . . . . . . . . . . 12 | |
37 | epel 4214 | . . . . . . . . . . . . 13 | |
38 | epel 4214 | . . . . . . . . . . . . 13 | |
39 | 37, 38 | orbi12i 753 | . . . . . . . . . . . 12 |
40 | 36, 39 | imbi12i 238 | . . . . . . . . . . 11 |
41 | 40 | 2ralbii 2443 | . . . . . . . . . 10 |
42 | 41 | ralbii 2441 | . . . . . . . . 9 |
43 | 35, 42 | mpbi 144 | . . . . . . . 8 |
44 | 43 | rspec3 2522 | . . . . . . 7 |
45 | 24, 31, 44 | vtocl 2740 | . . . . . 6 |
46 | 16, 23, 45 | vtocl 2740 | . . . . 5 |
47 | 2, 15, 46 | vtocl 2740 | . . . 4 |
48 | 1, 4, 8, 47 | mp3an 1315 | . . 3 |
49 | 2 | elsn 3543 | . . . . 5 |
50 | ordtriexmidlem2 4436 | . . . . 5 | |
51 | 49, 50 | sylbi 120 | . . . 4 |
52 | elirr 4456 | . . . . . . 7 | |
53 | elrabi 2837 | . . . . . . 7 | |
54 | 52, 53 | mto 651 | . . . . . 6 |
55 | elsuci 4325 | . . . . . . 7 | |
56 | 55 | ord 713 | . . . . . 6 |
57 | 54, 56 | mpi 15 | . . . . 5 |
58 | 0ex 4055 | . . . . . . 7 | |
59 | biidd 171 | . . . . . . 7 | |
60 | 58, 59 | rabsnt 3598 | . . . . . 6 |
61 | 60 | eqcoms 2142 | . . . . 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 2416 crab 2420 c0 3363 csn 3527 class class class wbr 3929 cep 4209 wpo 4216 wor 4217 con0 4285 csuc 4287 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-tr 4027 df-eprel 4211 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: (None) |
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