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Mirrors > Home > ILE Home > Th. List > ontri2orexmidim | Unicode version |
Description: Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4555. (Contributed by Jim Kingdon, 26-Aug-2024.) |
Ref | Expression |
---|---|
ontri2orexmidim | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2or2exmidlem 4510 | . . . . 5 | |
2 | suc0 4396 | . . . . . 6 | |
3 | 0elon 4377 | . . . . . . 7 | |
4 | 3 | onsuci 4500 | . . . . . 6 |
5 | 2, 4 | eqeltrri 2244 | . . . . 5 |
6 | sseq1 3170 | . . . . . . 7 | |
7 | sseq2 3171 | . . . . . . 7 | |
8 | 6, 7 | orbi12d 788 | . . . . . 6 |
9 | sseq2 3171 | . . . . . . 7 | |
10 | sseq1 3170 | . . . . . . 7 | |
11 | 9, 10 | orbi12d 788 | . . . . . 6 |
12 | 8, 11 | rspc2va 2848 | . . . . 5 |
13 | 1, 5, 12 | mpanl12 434 | . . . 4 |
14 | 5 | onirri 4527 | . . . . . 6 |
15 | simpl 108 | . . . . . . . 8 | |
16 | simpr 109 | . . . . . . . . 9 | |
17 | p0ex 4174 | . . . . . . . . . . 11 | |
18 | 17 | prid2 3690 | . . . . . . . . . 10 |
19 | biidd 171 | . . . . . . . . . . 11 | |
20 | 19 | elrab3 2887 | . . . . . . . . . 10 |
21 | 18, 20 | ax-mp 5 | . . . . . . . . 9 |
22 | 16, 21 | sylibr 133 | . . . . . . . 8 |
23 | 15, 22 | sseldd 3148 | . . . . . . 7 |
24 | 23 | ex 114 | . . . . . 6 |
25 | 14, 24 | mtoi 659 | . . . . 5 |
26 | snssg 3716 | . . . . . . 7 | |
27 | 3, 26 | ax-mp 5 | . . . . . 6 |
28 | 0ex 4116 | . . . . . . . 8 | |
29 | 28 | prid1 3689 | . . . . . . 7 |
30 | biidd 171 | . . . . . . . 8 | |
31 | 30 | elrab3 2887 | . . . . . . 7 |
32 | 29, 31 | ax-mp 5 | . . . . . 6 |
33 | 27, 32 | sylbb1 136 | . . . . 5 |
34 | 25, 33 | orim12i 754 | . . . 4 |
35 | 13, 34 | syl 14 | . . 3 |
36 | 35 | orcomd 724 | . 2 |
37 | df-dc 830 | . 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 703 DECID wdc 829 wceq 1348 wcel 2141 wral 2448 crab 2452 wss 3121 c0 3414 csn 3583 cpr 3584 con0 4348 csuc 4350 |
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-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 |
This theorem is referenced by: exmidontri2or 7220 |
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