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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 2231 | . . . . . . 7 | |
2 | equequ1 1705 | . . . . . . 7 | |
3 | eleq2 2234 | . . . . . . 7 | |
4 | 1, 2, 3 | 3orbi123d 1306 | . . . . . 6 |
5 | 4 | ralbidv 2470 | . . . . 5 |
6 | 5 | imbi2d 229 | . . . 4 EXMID EXMID |
7 | simplll 528 | . . . . . . . 8 EXMID EXMID | |
8 | simpr 109 | . . . . . . . 8 EXMID EXMID | |
9 | simplr 525 | . . . . . . . 8 EXMID EXMID EXMID | |
10 | simpllr 529 | . . . . . . . . 9 EXMID EXMID EXMID | |
11 | pm2.27 40 | . . . . . . . . . . 11 EXMID EXMID | |
12 | 11 | ralimdv 2538 | . . . . . . . . . 10 EXMID EXMID |
13 | 12 | ad2antlr 486 | . . . . . . . . 9 EXMID EXMID EXMID |
14 | 10, 13 | mpd 13 | . . . . . . . 8 EXMID EXMID |
15 | 7, 8, 9, 14 | exmidontriimlem4 7201 | . . . . . . 7 EXMID EXMID |
16 | 15 | ralrimiva 2543 | . . . . . 6 EXMID EXMID |
17 | eleq2 2234 | . . . . . . . 8 | |
18 | equequ2 1706 | . . . . . . . 8 | |
19 | eleq1w 2231 | . . . . . . . 8 | |
20 | 17, 18, 19 | 3orbi123d 1306 | . . . . . . 7 |
21 | 20 | cbvralv 2696 | . . . . . 6 |
22 | 16, 21 | sylib 121 | . . . . 5 EXMID EXMID |
23 | 22 | exp31 362 | . . . 4 EXMID EXMID |
24 | 6, 23 | tfis2 4569 | . . 3 EXMID |
25 | 24 | impcom 124 | . 2 EXMID |
26 | 25 | ralrimiva 2543 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3o 972 wcel 2141 wral 2448 EXMIDwem 4180 con0 4348 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-uni 3797 df-tr 4088 df-exmid 4181 df-iord 4351 df-on 4353 |
This theorem is referenced by: exmidontri 7216 onntri51 7217 exmidontri2or 7220 |
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