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Mirrors > Home > ILE Home > Th. List > ordtriexmid | Unicode version |
Description: Ordinal trichotomy
implies the law of the excluded middle (that is,
decidability of an arbitrary proposition).
This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.) |
Ref | Expression |
---|---|
ordtriexmid.1 |
Ref | Expression |
---|---|
ordtriexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3228 | . . . 4 | |
2 | ordtriexmidlem 4245 | . . . . . 6 | |
3 | eleq1 2100 | . . . . . . . 8 | |
4 | eqeq1 2046 | . . . . . . . 8 | |
5 | eleq2 2101 | . . . . . . . 8 | |
6 | 3, 4, 5 | 3orbi123d 1206 | . . . . . . 7 |
7 | 0elon 4129 | . . . . . . . 8 | |
8 | 0ex 3884 | . . . . . . . . 9 | |
9 | eleq1 2100 | . . . . . . . . . . 11 | |
10 | 9 | anbi2d 437 | . . . . . . . . . 10 |
11 | eleq2 2101 | . . . . . . . . . . 11 | |
12 | eqeq2 2049 | . . . . . . . . . . 11 | |
13 | eleq1 2100 | . . . . . . . . . . 11 | |
14 | 11, 12, 13 | 3orbi123d 1206 | . . . . . . . . . 10 |
15 | 10, 14 | imbi12d 223 | . . . . . . . . 9 |
16 | ordtriexmid.1 | . . . . . . . . . 10 | |
17 | 16 | rspec2 2408 | . . . . . . . . 9 |
18 | 8, 15, 17 | vtocl 2608 | . . . . . . . 8 |
19 | 7, 18 | mpan2 401 | . . . . . . 7 |
20 | 6, 19 | vtoclga 2619 | . . . . . 6 |
21 | 2, 20 | ax-mp 7 | . . . . 5 |
22 | 3orass 888 | . . . . 5 | |
23 | 21, 22 | mpbi 133 | . . . 4 |
24 | 1, 23 | mtpor 1316 | . . 3 |
25 | ordtriexmidlem2 4246 | . . . 4 | |
26 | 8 | snid 3402 | . . . . . 6 |
27 | biidd 161 | . . . . . . 7 | |
28 | 27 | elrab3 2699 | . . . . . 6 |
29 | 26, 28 | ax-mp 7 | . . . . 5 |
30 | 29 | biimpi 113 | . . . 4 |
31 | 25, 30 | orim12i 676 | . . 3 |
32 | 24, 31 | ax-mp 7 | . 2 |
33 | orcom 647 | . 2 | |
34 | 32, 33 | mpbir 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3o 884 wceq 1243 wcel 1393 wral 2306 crab 2310 c0 3224 csn 3375 con0 4100 |
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-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 |
This theorem depends on definitions: df-bi 110 df-3or 886 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-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-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 df-suc 4108 |
This theorem is referenced by: (None) |
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