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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 |
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Ref | Expression |
---|---|
ordtri2or2exmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2or2exmid.1 |
. . . 4
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2 | ordtri2or2exmidlem 4399 |
. . . . 5
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3 | suc0 4291 |
. . . . . 6
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4 | 0elon 4272 |
. . . . . . 7
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5 | 4 | onsuci 4390 |
. . . . . 6
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6 | 3, 5 | eqeltrri 2186 |
. . . . 5
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7 | sseq1 3084 |
. . . . . . 7
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8 | sseq2 3085 |
. . . . . . 7
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9 | 7, 8 | orbi12d 765 |
. . . . . 6
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10 | sseq2 3085 |
. . . . . . 7
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11 | sseq1 3084 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | orbi12d 765 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 9, 12 | rspc2va 2771 |
. . . . 5
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14 | 2, 6, 13 | mpanl12 430 |
. . . 4
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15 | 1, 14 | ax-mp 7 |
. . 3
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16 | elirr 4414 |
. . . . 5
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17 | simpl 108 |
. . . . . . 7
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18 | simpr 109 |
. . . . . . . 8
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19 | p0ex 4070 |
. . . . . . . . . 10
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20 | 19 | prid2 3594 |
. . . . . . . . 9
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21 | biidd 171 |
. . . . . . . . . 10
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22 | 21 | elrab3 2808 |
. . . . . . . . 9
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23 | 20, 22 | ax-mp 7 |
. . . . . . . 8
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24 | 18, 23 | sylibr 133 |
. . . . . . 7
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25 | 17, 24 | sseldd 3062 |
. . . . . 6
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26 | 25 | ex 114 |
. . . . 5
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27 | 16, 26 | mtoi 636 |
. . . 4
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28 | snssg 3620 |
. . . . . 6
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29 | 4, 28 | ax-mp 7 |
. . . . 5
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30 | 0ex 4013 |
. . . . . . . 8
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31 | 30 | prid1 3593 |
. . . . . . 7
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32 | biidd 171 |
. . . . . . . 8
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33 | 32 | elrab3 2808 |
. . . . . . 7
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34 | 31, 33 | ax-mp 7 |
. . . . . 6
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35 | 34 | biimpi 119 |
. . . . 5
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36 | 29, 35 | sylbir 134 |
. . . 4
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37 | 27, 36 | orim12i 731 |
. . 3
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38 | 15, 37 | ax-mp 7 |
. 2
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39 | orcom 700 |
. 2
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40 | 38, 39 | mpbi 144 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-tr 3985 df-iord 4246 df-on 4248 df-suc 4251 |
This theorem is referenced by: onintexmid 4445 |
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