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Mirrors > Home > ILE Home > Th. List > ordtriexmid | Unicode version |
Description: Ordinal trichotomy
implies the law of the excluded middle (that is,
decidability of an arbitrary proposition).
This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.) |
Ref | Expression |
---|---|
ordtriexmid.1 |
Ref | Expression |
---|---|
ordtriexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3337 | . . . 4 | |
2 | ordtriexmidlem 4405 | . . . . . 6 | |
3 | eleq1 2180 | . . . . . . . 8 | |
4 | eqeq1 2124 | . . . . . . . 8 | |
5 | eleq2 2181 | . . . . . . . 8 | |
6 | 3, 4, 5 | 3orbi123d 1274 | . . . . . . 7 |
7 | 0elon 4284 | . . . . . . . 8 | |
8 | 0ex 4025 | . . . . . . . . 9 | |
9 | eleq1 2180 | . . . . . . . . . . 11 | |
10 | 9 | anbi2d 459 | . . . . . . . . . 10 |
11 | eleq2 2181 | . . . . . . . . . . 11 | |
12 | eqeq2 2127 | . . . . . . . . . . 11 | |
13 | eleq1 2180 | . . . . . . . . . . 11 | |
14 | 11, 12, 13 | 3orbi123d 1274 | . . . . . . . . . 10 |
15 | 10, 14 | imbi12d 233 | . . . . . . . . 9 |
16 | ordtriexmid.1 | . . . . . . . . . 10 | |
17 | 16 | rspec2 2498 | . . . . . . . . 9 |
18 | 8, 15, 17 | vtocl 2714 | . . . . . . . 8 |
19 | 7, 18 | mpan2 421 | . . . . . . 7 |
20 | 6, 19 | vtoclga 2726 | . . . . . 6 |
21 | 2, 20 | ax-mp 5 | . . . . 5 |
22 | 3orass 950 | . . . . 5 | |
23 | 21, 22 | mpbi 144 | . . . 4 |
24 | 1, 23 | mtpor 1388 | . . 3 |
25 | ordtriexmidlem2 4406 | . . . 4 | |
26 | 8 | snid 3526 | . . . . . 6 |
27 | biidd 171 | . . . . . . 7 | |
28 | 27 | elrab3 2814 | . . . . . 6 |
29 | 26, 28 | ax-mp 5 | . . . . 5 |
30 | 29 | biimpi 119 | . . . 4 |
31 | 25, 30 | orim12i 733 | . . 3 |
32 | 24, 31 | ax-mp 5 | . 2 |
33 | orcom 702 | . 2 | |
34 | 32, 33 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 wral 2393 crab 2397 c0 3333 csn 3497 con0 4255 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 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-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 |
This theorem is referenced by: (None) |
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