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Mirrors > Home > ILE Home > Th. List > fimax2gtri | Unicode version |
Description: A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.) |
Ref | Expression |
---|---|
fimax2gtri.po |
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fimax2gtri.tri |
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fimax2gtri.fin |
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fimax2gtri.n0 |
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Ref | Expression |
---|---|
fimax2gtri |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 2629 |
. . 3
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2 | 1 | rexbidv 2439 |
. 2
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3 | raleq 2629 |
. . 3
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4 | 3 | rexbidv 2439 |
. 2
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5 | raleq 2629 |
. . 3
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6 | 5 | rexbidv 2439 |
. 2
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7 | raleq 2629 |
. . 3
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8 | 7 | rexbidv 2439 |
. 2
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9 | fimax2gtri.n0 |
. . . . 5
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10 | fimax2gtri.fin |
. . . . . 6
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11 | fin0 6787 |
. . . . . 6
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12 | 10, 11 | syl 14 |
. . . . 5
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13 | 9, 12 | mpbid 146 |
. . . 4
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14 | ral0 3469 |
. . . . . 6
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15 | 14 | biantru 300 |
. . . . 5
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16 | 15 | exbii 1585 |
. . . 4
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17 | 13, 16 | sylib 121 |
. . 3
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18 | df-rex 2423 |
. . 3
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19 | 17, 18 | sylibr 133 |
. 2
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20 | breq1 3940 |
. . . . . 6
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21 | 20 | notbid 657 |
. . . . 5
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22 | 21 | ralbidv 2438 |
. . . 4
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23 | 22 | cbvrexv 2658 |
. . 3
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24 | fimax2gtri.po |
. . . . . . 7
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25 | 24 | ad4antr 486 |
. . . . . 6
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26 | fimax2gtri.tri |
. . . . . . 7
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27 | 26 | ad4antr 486 |
. . . . . 6
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28 | 10 | ad4antr 486 |
. . . . . 6
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29 | 9 | ad4antr 486 |
. . . . . 6
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30 | simp-4r 532 |
. . . . . 6
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31 | simprl 521 |
. . . . . . 7
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32 | 31 | ad2antrr 480 |
. . . . . 6
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33 | simplr 520 |
. . . . . 6
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34 | simprr 522 |
. . . . . . . 8
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35 | 34 | ad2antrr 480 |
. . . . . . 7
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36 | 35 | eldifad 3087 |
. . . . . 6
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37 | 35 | eldifbd 3088 |
. . . . . 6
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38 | simpr 109 |
. . . . . 6
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39 | 25, 27, 28, 29, 30, 32, 33, 36, 37, 38 | fimax2gtrilemstep 6802 |
. . . . 5
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40 | 39 | ex 114 |
. . . 4
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41 | 40 | rexlimdva 2552 |
. . 3
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42 | 23, 41 | syl5bi 151 |
. 2
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43 | 2, 4, 6, 8, 19, 42, 10 | findcard2sd 6794 |
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-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-fin 6645 |
This theorem is referenced by: fimaxq 10605 |
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