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Mirrors > Home > ILE Home > Th. List > exmidontriim | Unicode version |
Description: Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.) |
Ref | Expression |
---|---|
exmidontriim | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2218 | . . . . . . 7 | |
2 | equequ1 1692 | . . . . . . 7 | |
3 | eleq2 2221 | . . . . . . 7 | |
4 | 1, 2, 3 | 3orbi123d 1293 | . . . . . 6 |
5 | 4 | ralbidv 2457 | . . . . 5 |
6 | 5 | imbi2d 229 | . . . 4 EXMID EXMID |
7 | simplll 523 | . . . . . . . 8 EXMID EXMID | |
8 | simpr 109 | . . . . . . . 8 EXMID EXMID | |
9 | simplr 520 | . . . . . . . 8 EXMID EXMID EXMID | |
10 | simpllr 524 | . . . . . . . . 9 EXMID EXMID EXMID | |
11 | pm2.27 40 | . . . . . . . . . . 11 EXMID EXMID | |
12 | 11 | ralimdv 2525 | . . . . . . . . . 10 EXMID EXMID |
13 | 12 | ad2antlr 481 | . . . . . . . . 9 EXMID EXMID EXMID |
14 | 10, 13 | mpd 13 | . . . . . . . 8 EXMID EXMID |
15 | 7, 8, 9, 14 | exmidontriimlem4 7160 | . . . . . . 7 EXMID EXMID |
16 | 15 | ralrimiva 2530 | . . . . . 6 EXMID EXMID |
17 | eleq2 2221 | . . . . . . . 8 | |
18 | equequ2 1693 | . . . . . . . 8 | |
19 | eleq1w 2218 | . . . . . . . 8 | |
20 | 17, 18, 19 | 3orbi123d 1293 | . . . . . . 7 |
21 | 20 | cbvralv 2680 | . . . . . 6 |
22 | 16, 21 | sylib 121 | . . . . 5 EXMID EXMID |
23 | 22 | exp31 362 | . . . 4 EXMID EXMID |
24 | 6, 23 | tfis2 4545 | . . 3 EXMID |
25 | 24 | impcom 124 | . 2 EXMID |
26 | 25 | ralrimiva 2530 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3o 962 wcel 2128 wral 2435 EXMIDwem 4156 con0 4324 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-uni 3774 df-tr 4064 df-exmid 4157 df-iord 4327 df-on 4329 |
This theorem is referenced by: exmidontri 7175 onntri51 7176 exmidontri2or 7179 |
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