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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 4351 |
. . . . 5
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3 | suc0 4249 |
. . . . . 6
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4 | 0elon 4230 |
. . . . . . 7
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5 | 4 | onsuci 4348 |
. . . . . 6
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6 | 3, 5 | eqeltrri 2162 |
. . . . 5
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7 | eleq1 2151 |
. . . . . . 7
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8 | sseq2 3051 |
. . . . . . 7
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9 | 7, 8 | orbi12d 743 |
. . . . . 6
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10 | eleq2 2152 |
. . . . . . 7
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11 | sseq1 3050 |
. . . . . . 7
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12 | 10, 11 | orbi12d 743 |
. . . . . 6
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13 | 9, 12 | rspc2va 2738 |
. . . . 5
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14 | 2, 6, 13 | mpanl12 428 |
. . . 4
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15 | 1, 14 | ax-mp 7 |
. . 3
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16 | elsni 3470 |
. . . . 5
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17 | ordtriexmidlem2 4352 |
. . . . 5
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18 | 16, 17 | syl 14 |
. . . 4
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19 | snssg 3581 |
. . . . . 6
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20 | 4, 19 | ax-mp 7 |
. . . . 5
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21 | 0ex 3974 |
. . . . . . . 8
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22 | 21 | snid 3481 |
. . . . . . 7
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23 | biidd 171 |
. . . . . . . 8
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24 | 23 | elrab3 2775 |
. . . . . . 7
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25 | 22, 24 | ax-mp 7 |
. . . . . 6
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26 | 25 | biimpi 119 |
. . . . 5
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27 | 20, 26 | sylbir 134 |
. . . 4
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28 | 18, 27 | orim12i 712 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 15, 28 | ax-mp 7 |
. 2
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30 | orcom 683 |
. 2
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31 | 29, 30 | 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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2624 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-uni 3662 df-tr 3945 df-iord 4204 df-on 4206 df-suc 4209 |
This theorem is referenced by: (None) |
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