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Mirrors > Home > ILE Home > Th. List > ordtri2orexmid | Unicode version |
Description: Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.) |
Ref | Expression |
---|---|
ordtri2orexmid.1 |
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Ref | Expression |
---|---|
ordtri2orexmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2orexmid.1 |
. . . 4
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2 | ordtriexmidlem 4208 |
. . . . 5
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3 | suc0 4114 |
. . . . . 6
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4 | 0elon 4095 |
. . . . . . 7
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5 | 4 | onsuci 4207 |
. . . . . 6
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6 | 3, 5 | eqeltrri 2108 |
. . . . 5
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7 | eleq1 2097 |
. . . . . . 7
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8 | sseq2 2961 |
. . . . . . 7
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9 | 7, 8 | orbi12d 706 |
. . . . . 6
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10 | eleq2 2098 |
. . . . . . 7
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11 | sseq1 2960 |
. . . . . . 7
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12 | 10, 11 | orbi12d 706 |
. . . . . 6
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13 | 9, 12 | rspc2va 2657 |
. . . . 5
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14 | 2, 6, 13 | mpanl12 412 |
. . . 4
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15 | 1, 14 | ax-mp 7 |
. . 3
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16 | elsni 3391 |
. . . . 5
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17 | ordtriexmidlem2 4209 |
. . . . 5
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18 | 16, 17 | syl 14 |
. . . 4
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19 | snssg 3491 |
. . . . . 6
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20 | 4, 19 | ax-mp 7 |
. . . . 5
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21 | 0ex 3875 |
. . . . . . . 8
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22 | 21 | snid 3394 |
. . . . . . 7
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23 | biidd 161 |
. . . . . . . 8
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24 | 23 | elrab3 2693 |
. . . . . . 7
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25 | 22, 24 | ax-mp 7 |
. . . . . 6
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26 | 25 | biimpi 113 |
. . . . 5
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27 | 20, 26 | sylbir 125 |
. . . 4
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28 | 18, 27 | orim12i 675 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 15, 28 | ax-mp 7 |
. 2
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30 | orcom 646 |
. 2
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31 | 29, 30 | mpbi 133 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-tr 3846 df-iord 4069 df-on 4071 df-suc 4074 |
This theorem is referenced by: (None) |
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