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Mirrors > Home > ILE Home > Th. List > ordtri2orexmid | Unicode version |
Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Ref | Expression |
---|---|
ordtri2orexmid.1 |
Ref | Expression |
---|---|
ordtri2orexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2orexmid.1 | . . . 4 | |
2 | ordtriexmidlem 4478 | . . . . 5 | |
3 | suc0 4371 | . . . . . 6 | |
4 | 0elon 4352 | . . . . . . 7 | |
5 | 4 | onsuci 4475 | . . . . . 6 |
6 | 3, 5 | eqeltrri 2231 | . . . . 5 |
7 | eleq1 2220 | . . . . . . 7 | |
8 | sseq2 3152 | . . . . . . 7 | |
9 | 7, 8 | orbi12d 783 | . . . . . 6 |
10 | eleq2 2221 | . . . . . . 7 | |
11 | sseq1 3151 | . . . . . . 7 | |
12 | 10, 11 | orbi12d 783 | . . . . . 6 |
13 | 9, 12 | rspc2va 2830 | . . . . 5 |
14 | 2, 6, 13 | mpanl12 433 | . . . 4 |
15 | 1, 14 | ax-mp 5 | . . 3 |
16 | elsni 3578 | . . . . 5 | |
17 | ordtriexmidlem2 4479 | . . . . 5 | |
18 | 16, 17 | syl 14 | . . . 4 |
19 | snssg 3692 | . . . . . 6 | |
20 | 4, 19 | ax-mp 5 | . . . . 5 |
21 | 0ex 4091 | . . . . . . . 8 | |
22 | 21 | snid 3591 | . . . . . . 7 |
23 | biidd 171 | . . . . . . . 8 | |
24 | 23 | elrab3 2869 | . . . . . . 7 |
25 | 22, 24 | ax-mp 5 | . . . . . 6 |
26 | 25 | biimpi 119 | . . . . 5 |
27 | 20, 26 | sylbir 134 | . . . 4 |
28 | 18, 27 | orim12i 749 | . . 3 |
29 | 15, 28 | ax-mp 5 | . 2 |
30 | orcom 718 | . 2 | |
31 | 29, 30 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wb 104 wo 698 wceq 1335 wcel 2128 wral 2435 crab 2439 wss 3102 c0 3394 csn 3560 con0 4323 csuc 4325 |
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-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 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-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 df-tr 4063 df-iord 4326 df-on 4328 df-suc 4331 |
This theorem is referenced by: (None) |
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