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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 4251 | . . . . 5 | |
3 | suc0 4148 | . . . . . 6 | |
4 | 0elon 4129 | . . . . . . 7 | |
5 | 4 | onsuci 4242 | . . . . . 6 |
6 | 3, 5 | eqeltrri 2111 | . . . . 5 |
7 | sseq1 2966 | . . . . . . 7 | |
8 | sseq2 2967 | . . . . . . 7 | |
9 | 7, 8 | orbi12d 707 | . . . . . 6 |
10 | sseq2 2967 | . . . . . . 7 | |
11 | sseq1 2966 | . . . . . . 7 | |
12 | 10, 11 | orbi12d 707 | . . . . . 6 |
13 | 9, 12 | rspc2va 2663 | . . . . 5 |
14 | 2, 6, 13 | mpanl12 412 | . . . 4 |
15 | 1, 14 | ax-mp 7 | . . 3 |
16 | elirr 4266 | . . . . 5 | |
17 | simpl 102 | . . . . . . 7 | |
18 | simpr 103 | . . . . . . . 8 | |
19 | p0ex 3939 | . . . . . . . . . 10 | |
20 | 19 | prid2 3477 | . . . . . . . . 9 |
21 | biidd 161 | . . . . . . . . . 10 | |
22 | 21 | elrab3 2699 | . . . . . . . . 9 |
23 | 20, 22 | ax-mp 7 | . . . . . . . 8 |
24 | 18, 23 | sylibr 137 | . . . . . . 7 |
25 | 17, 24 | sseldd 2946 | . . . . . 6 |
26 | 25 | ex 108 | . . . . 5 |
27 | 16, 26 | mtoi 590 | . . . 4 |
28 | snssg 3500 | . . . . . 6 | |
29 | 4, 28 | ax-mp 7 | . . . . 5 |
30 | 0ex 3884 | . . . . . . . 8 | |
31 | 30 | prid1 3476 | . . . . . . 7 |
32 | biidd 161 | . . . . . . . 8 | |
33 | 32 | elrab3 2699 | . . . . . . 7 |
34 | 31, 33 | ax-mp 7 | . . . . . 6 |
35 | 34 | biimpi 113 | . . . . 5 |
36 | 29, 35 | sylbir 125 | . . . 4 |
37 | 27, 36 | orim12i 676 | . . 3 |
38 | 15, 37 | ax-mp 7 | . 2 |
39 | orcom 647 | . 2 | |
40 | 38, 39 | mpbi 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 wral 2306 crab 2310 wss 2917 c0 3224 csn 3375 cpr 3376 con0 4100 csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: onintexmid 4297 |
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