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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 4443 |
. . . . 5
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3 | suc0 4341 |
. . . . . 6
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4 | 0elon 4322 |
. . . . . . 7
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5 | 4 | onsuci 4440 |
. . . . . 6
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6 | 3, 5 | eqeltrri 2214 |
. . . . 5
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7 | eleq1 2203 |
. . . . . . 7
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8 | sseq2 3126 |
. . . . . . 7
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9 | 7, 8 | orbi12d 783 |
. . . . . 6
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10 | eleq2 2204 |
. . . . . . 7
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11 | sseq1 3125 |
. . . . . . 7
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12 | 10, 11 | orbi12d 783 |
. . . . . 6
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13 | 9, 12 | rspc2va 2807 |
. . . . 5
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14 | 2, 6, 13 | mpanl12 433 |
. . . 4
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15 | 1, 14 | ax-mp 5 |
. . 3
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16 | elsni 3550 |
. . . . 5
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17 | ordtriexmidlem2 4444 |
. . . . 5
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18 | 16, 17 | syl 14 |
. . . 4
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19 | snssg 3664 |
. . . . . 6
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20 | 4, 19 | ax-mp 5 |
. . . . 5
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21 | 0ex 4063 |
. . . . . . . 8
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22 | 21 | snid 3563 |
. . . . . . 7
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23 | biidd 171 |
. . . . . . . 8
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24 | 23 | elrab3 2845 |
. . . . . . 7
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25 | 22, 24 | ax-mp 5 |
. . . . . 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 749 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 15, 28 | ax-mp 5 |
. 2
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30 | orcom 718 |
. 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-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 |
This theorem is referenced by: (None) |
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