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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 4543 |
. . . . 5
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3 | suc0 4429 |
. . . . . 6
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4 | 0elon 4410 |
. . . . . . 7
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5 | 4 | onsuci 4533 |
. . . . . 6
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6 | 3, 5 | eqeltrri 2263 |
. . . . 5
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7 | sseq1 3193 |
. . . . . . 7
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8 | sseq2 3194 |
. . . . . . 7
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9 | 7, 8 | orbi12d 794 |
. . . . . 6
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10 | sseq2 3194 |
. . . . . . 7
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11 | sseq1 3193 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | orbi12d 794 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 9, 12 | rspc2va 2870 |
. . . . 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 | elirr 4558 |
. . . . 5
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17 | simpl 109 |
. . . . . . 7
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18 | simpr 110 |
. . . . . . . 8
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19 | p0ex 4206 |
. . . . . . . . . 10
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20 | 19 | prid2 3714 |
. . . . . . . . 9
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21 | biidd 172 |
. . . . . . . . . 10
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22 | 21 | elrab3 2909 |
. . . . . . . . 9
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23 | 20, 22 | ax-mp 5 |
. . . . . . . 8
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24 | 18, 23 | sylibr 134 |
. . . . . . 7
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25 | 17, 24 | sseldd 3171 |
. . . . . 6
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26 | 25 | ex 115 |
. . . . 5
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27 | 16, 26 | mtoi 665 |
. . . 4
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28 | snssg 3741 |
. . . . . 6
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29 | 4, 28 | ax-mp 5 |
. . . . 5
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30 | 0ex 4145 |
. . . . . . . 8
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31 | 30 | prid1 3713 |
. . . . . . 7
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32 | biidd 172 |
. . . . . . . 8
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33 | 32 | elrab3 2909 |
. . . . . . 7
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34 | 31, 33 | ax-mp 5 |
. . . . . 6
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35 | 34 | biimpi 120 |
. . . . 5
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36 | 29, 35 | sylbir 135 |
. . . 4
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37 | 27, 36 | orim12i 760 |
. . 3
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38 | 15, 37 | ax-mp 5 |
. 2
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39 | orcom 729 |
. 2
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40 | 38, 39 | 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-tr 4117 df-iord 4384 df-on 4386 df-suc 4389 |
This theorem is referenced by: onintexmid 4590 |
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