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Mirrors > Home > ILE Home > Th. List > ontri2orexmidim | Unicode version |
Description: Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4531. (Contributed by Jim Kingdon, 26-Aug-2024.) |
Ref | Expression |
---|---|
ontri2orexmidim | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2or2exmidlem 4486 | . . . . 5 | |
2 | suc0 4372 | . . . . . 6 | |
3 | 0elon 4353 | . . . . . . 7 | |
4 | 3 | onsuci 4476 | . . . . . 6 |
5 | 2, 4 | eqeltrri 2231 | . . . . 5 |
6 | sseq1 3151 | . . . . . . 7 | |
7 | sseq2 3152 | . . . . . . 7 | |
8 | 6, 7 | orbi12d 783 | . . . . . 6 |
9 | sseq2 3152 | . . . . . . 7 | |
10 | sseq1 3151 | . . . . . . 7 | |
11 | 9, 10 | orbi12d 783 | . . . . . 6 |
12 | 8, 11 | rspc2va 2830 | . . . . 5 |
13 | 1, 5, 12 | mpanl12 433 | . . . 4 |
14 | 5 | onirri 4503 | . . . . . 6 |
15 | simpl 108 | . . . . . . . 8 | |
16 | simpr 109 | . . . . . . . . 9 | |
17 | p0ex 4150 | . . . . . . . . . . 11 | |
18 | 17 | prid2 3667 | . . . . . . . . . 10 |
19 | biidd 171 | . . . . . . . . . . 11 | |
20 | 19 | elrab3 2869 | . . . . . . . . . 10 |
21 | 18, 20 | ax-mp 5 | . . . . . . . . 9 |
22 | 16, 21 | sylibr 133 | . . . . . . . 8 |
23 | 15, 22 | sseldd 3129 | . . . . . . 7 |
24 | 23 | ex 114 | . . . . . 6 |
25 | 14, 24 | mtoi 654 | . . . . 5 |
26 | snssg 3693 | . . . . . . 7 | |
27 | 3, 26 | ax-mp 5 | . . . . . 6 |
28 | 0ex 4092 | . . . . . . . 8 | |
29 | 28 | prid1 3666 | . . . . . . 7 |
30 | biidd 171 | . . . . . . . 8 | |
31 | 30 | elrab3 2869 | . . . . . . 7 |
32 | 29, 31 | ax-mp 5 | . . . . . 6 |
33 | 27, 32 | sylbb1 136 | . . . . 5 |
34 | 25, 33 | orim12i 749 | . . . 4 |
35 | 13, 34 | syl 14 | . . 3 |
36 | 35 | orcomd 719 | . 2 |
37 | df-dc 821 | . 2 DECID | |
38 | 36, 37 | sylibr 133 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 820 wceq 1335 wcel 2128 wral 2435 crab 2439 wss 3102 c0 3394 csn 3560 cpr 3561 con0 4324 csuc 4326 |
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 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-dc 821 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-ne 2328 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 3774 df-tr 4064 df-iord 4327 df-on 4329 df-suc 4332 |
This theorem is referenced by: exmidontri2or 7179 |
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