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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 3367 | . . . 4 | |
2 | ordtriexmidlem 4435 | . . . . . 6 | |
3 | eleq1 2202 | . . . . . . . 8 | |
4 | eqeq1 2146 | . . . . . . . 8 | |
5 | eleq2 2203 | . . . . . . . 8 | |
6 | 3, 4, 5 | 3orbi123d 1289 | . . . . . . 7 |
7 | 0elon 4314 | . . . . . . . 8 | |
8 | 0ex 4055 | . . . . . . . . 9 | |
9 | eleq1 2202 | . . . . . . . . . . 11 | |
10 | 9 | anbi2d 459 | . . . . . . . . . 10 |
11 | eleq2 2203 | . . . . . . . . . . 11 | |
12 | eqeq2 2149 | . . . . . . . . . . 11 | |
13 | eleq1 2202 | . . . . . . . . . . 11 | |
14 | 11, 12, 13 | 3orbi123d 1289 | . . . . . . . . . 10 |
15 | 10, 14 | imbi12d 233 | . . . . . . . . 9 |
16 | ordtriexmid.1 | . . . . . . . . . 10 | |
17 | 16 | rspec2 2521 | . . . . . . . . 9 |
18 | 8, 15, 17 | vtocl 2740 | . . . . . . . 8 |
19 | 7, 18 | mpan2 421 | . . . . . . 7 |
20 | 6, 19 | vtoclga 2752 | . . . . . 6 |
21 | 2, 20 | ax-mp 5 | . . . . 5 |
22 | 3orass 965 | . . . . 5 | |
23 | 21, 22 | mpbi 144 | . . . 4 |
24 | 1, 23 | mtpor 1403 | . . 3 |
25 | ordtriexmidlem2 4436 | . . . 4 | |
26 | 8 | snid 3556 | . . . . . 6 |
27 | biidd 171 | . . . . . . 7 | |
28 | 27 | elrab3 2841 | . . . . . 6 |
29 | 26, 28 | ax-mp 5 | . . . . 5 |
30 | 29 | biimpi 119 | . . . 4 |
31 | 25, 30 | orim12i 748 | . . 3 |
32 | 24, 31 | ax-mp 5 | . 2 |
33 | orcom 717 | . 2 | |
34 | 32, 33 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3o 961 wceq 1331 wcel 1480 wral 2416 crab 2420 c0 3363 csn 3527 con0 4285 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 |
This theorem is referenced by: (None) |
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