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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 3255 | . . . 4 | |
2 | ordtriexmidlem 4263 | . . . . . 6 | |
3 | eleq1 2141 | . . . . . . . 8 | |
4 | eqeq1 2087 | . . . . . . . 8 | |
5 | eleq2 2142 | . . . . . . . 8 | |
6 | 3, 4, 5 | 3orbi123d 1242 | . . . . . . 7 |
7 | 0elon 4147 | . . . . . . . 8 | |
8 | 0ex 3905 | . . . . . . . . 9 | |
9 | eleq1 2141 | . . . . . . . . . . 11 | |
10 | 9 | anbi2d 451 | . . . . . . . . . 10 |
11 | eleq2 2142 | . . . . . . . . . . 11 | |
12 | eqeq2 2090 | . . . . . . . . . . 11 | |
13 | eleq1 2141 | . . . . . . . . . . 11 | |
14 | 11, 12, 13 | 3orbi123d 1242 | . . . . . . . . . 10 |
15 | 10, 14 | imbi12d 232 | . . . . . . . . 9 |
16 | ordtriexmid.1 | . . . . . . . . . 10 | |
17 | 16 | rspec2 2450 | . . . . . . . . 9 |
18 | 8, 15, 17 | vtocl 2653 | . . . . . . . 8 |
19 | 7, 18 | mpan2 415 | . . . . . . 7 |
20 | 6, 19 | vtoclga 2664 | . . . . . 6 |
21 | 2, 20 | ax-mp 7 | . . . . 5 |
22 | 3orass 922 | . . . . 5 | |
23 | 21, 22 | mpbi 143 | . . . 4 |
24 | 1, 23 | mtpor 1356 | . . 3 |
25 | ordtriexmidlem2 4264 | . . . 4 | |
26 | 8 | snid 3425 | . . . . . 6 |
27 | biidd 170 | . . . . . . 7 | |
28 | 27 | elrab3 2750 | . . . . . 6 |
29 | 26, 28 | ax-mp 7 | . . . . 5 |
30 | 29 | biimpi 118 | . . . 4 |
31 | 25, 30 | orim12i 708 | . . 3 |
32 | 24, 31 | ax-mp 7 | . 2 |
33 | orcom 679 | . 2 | |
34 | 32, 33 | mpbir 144 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3o 918 wceq 1284 wcel 1433 wral 2348 crab 2352 c0 3251 csn 3398 con0 4118 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 |
This theorem is referenced by: (None) |
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