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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 4530 |
. . . . 5
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3 | suc0 4423 |
. . . . . 6
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4 | 0elon 4404 |
. . . . . . 7
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5 | 4 | onsuci 4527 |
. . . . . 6
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6 | 3, 5 | eqeltrri 2261 |
. . . . 5
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7 | eleq1 2250 |
. . . . . . 7
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8 | sseq2 3191 |
. . . . . . 7
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9 | 7, 8 | orbi12d 794 |
. . . . . 6
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10 | eleq2 2251 |
. . . . . . 7
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11 | sseq1 3190 |
. . . . . . 7
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12 | 10, 11 | orbi12d 794 |
. . . . . 6
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13 | 9, 12 | rspc2va 2867 |
. . . . 5
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14 | 2, 6, 13 | mpanl12 436 |
. . . 4
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15 | 1, 14 | ax-mp 5 |
. . 3
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16 | elsni 3622 |
. . . . 5
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17 | ordtriexmidlem2 4531 |
. . . . 5
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18 | 16, 17 | syl 14 |
. . . 4
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19 | snssg 3738 |
. . . . . 6
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20 | 4, 19 | ax-mp 5 |
. . . . 5
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21 | 0ex 4142 |
. . . . . . . 8
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22 | 21 | snid 3635 |
. . . . . . 7
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23 | biidd 172 |
. . . . . . . 8
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24 | 23 | elrab3 2906 |
. . . . . . 7
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25 | 22, 24 | ax-mp 5 |
. . . . . 6
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26 | 25 | biimpi 120 |
. . . . 5
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27 | 20, 26 | sylbir 135 |
. . . 4
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28 | 18, 27 | orim12i 760 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 15, 28 | ax-mp 5 |
. 2
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30 | orcom 729 |
. 2
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31 | 29, 30 | mpbi 145 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 df-tr 4114 df-iord 4378 df-on 4380 df-suc 4383 |
This theorem is referenced by: (None) |
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