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Mirrors > Home > ILE Home > Th. List > ordtri2or2exmid | Unicode version |
Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Ref | Expression |
---|---|
ordtri2or2exmid.1 |
Ref | Expression |
---|---|
ordtri2or2exmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2or2exmid.1 | . . . 4 | |
2 | ordtri2or2exmidlem 4497 | . . . . 5 | |
3 | suc0 4383 | . . . . . 6 | |
4 | 0elon 4364 | . . . . . . 7 | |
5 | 4 | onsuci 4487 | . . . . . 6 |
6 | 3, 5 | eqeltrri 2238 | . . . . 5 |
7 | sseq1 3160 | . . . . . . 7 | |
8 | sseq2 3161 | . . . . . . 7 | |
9 | 7, 8 | orbi12d 783 | . . . . . 6 |
10 | sseq2 3161 | . . . . . . 7 | |
11 | sseq1 3160 | . . . . . . 7 | |
12 | 10, 11 | orbi12d 783 | . . . . . 6 |
13 | 9, 12 | rspc2va 2839 | . . . . 5 |
14 | 2, 6, 13 | mpanl12 433 | . . . 4 |
15 | 1, 14 | ax-mp 5 | . . 3 |
16 | elirr 4512 | . . . . 5 | |
17 | simpl 108 | . . . . . . 7 | |
18 | simpr 109 | . . . . . . . 8 | |
19 | p0ex 4161 | . . . . . . . . . 10 | |
20 | 19 | prid2 3677 | . . . . . . . . 9 |
21 | biidd 171 | . . . . . . . . . 10 | |
22 | 21 | elrab3 2878 | . . . . . . . . 9 |
23 | 20, 22 | ax-mp 5 | . . . . . . . 8 |
24 | 18, 23 | sylibr 133 | . . . . . . 7 |
25 | 17, 24 | sseldd 3138 | . . . . . 6 |
26 | 25 | ex 114 | . . . . 5 |
27 | 16, 26 | mtoi 654 | . . . 4 |
28 | snssg 3703 | . . . . . 6 | |
29 | 4, 28 | ax-mp 5 | . . . . 5 |
30 | 0ex 4103 | . . . . . . . 8 | |
31 | 30 | prid1 3676 | . . . . . . 7 |
32 | biidd 171 | . . . . . . . 8 | |
33 | 32 | elrab3 2878 | . . . . . . 7 |
34 | 31, 33 | ax-mp 5 | . . . . . 6 |
35 | 34 | biimpi 119 | . . . . 5 |
36 | 29, 35 | sylbir 134 | . . . 4 |
37 | 27, 36 | orim12i 749 | . . 3 |
38 | 15, 37 | ax-mp 5 | . 2 |
39 | orcom 718 | . 2 | |
40 | 38, 39 | mpbi 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wral 2442 crab 2446 wss 3111 c0 3404 csn 3570 cpr 3571 con0 4335 csuc 4337 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 |
This theorem is referenced by: onintexmid 4544 |
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