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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 2673 |
. . 3
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2 | 1 | rexbidv 2478 |
. 2
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3 | raleq 2673 |
. . 3
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4 | 3 | rexbidv 2478 |
. 2
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5 | raleq 2673 |
. . 3
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6 | 5 | rexbidv 2478 |
. 2
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7 | raleq 2673 |
. . 3
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8 | 7 | rexbidv 2478 |
. 2
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9 | fimax2gtri.n0 |
. . . . 5
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10 | fimax2gtri.fin |
. . . . . 6
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11 | fin0 6885 |
. . . . . 6
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12 | 10, 11 | syl 14 |
. . . . 5
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13 | 9, 12 | mpbid 147 |
. . . 4
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14 | ral0 3525 |
. . . . . 6
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15 | 14 | biantru 302 |
. . . . 5
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16 | 15 | exbii 1605 |
. . . 4
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17 | 13, 16 | sylib 122 |
. . 3
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18 | df-rex 2461 |
. . 3
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19 | 17, 18 | sylibr 134 |
. 2
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20 | breq1 4007 |
. . . . . 6
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21 | 20 | notbid 667 |
. . . . 5
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22 | 21 | ralbidv 2477 |
. . . 4
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23 | 22 | cbvrexv 2705 |
. . 3
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24 | fimax2gtri.po |
. . . . . . 7
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25 | 24 | ad4antr 494 |
. . . . . 6
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26 | fimax2gtri.tri |
. . . . . . 7
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27 | 26 | ad4antr 494 |
. . . . . 6
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28 | 10 | ad4antr 494 |
. . . . . 6
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29 | 9 | ad4antr 494 |
. . . . . 6
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30 | simp-4r 542 |
. . . . . 6
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31 | simprl 529 |
. . . . . . 7
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32 | 31 | ad2antrr 488 |
. . . . . 6
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33 | simplr 528 |
. . . . . 6
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34 | simprr 531 |
. . . . . . . 8
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35 | 34 | ad2antrr 488 |
. . . . . . 7
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36 | 35 | eldifad 3141 |
. . . . . 6
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37 | 35 | eldifbd 3142 |
. . . . . 6
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38 | simpr 110 |
. . . . . 6
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39 | 25, 27, 28, 29, 30, 32, 33, 36, 37, 38 | fimax2gtrilemstep 6900 |
. . . . 5
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40 | 39 | ex 115 |
. . . 4
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41 | 40 | rexlimdva 2594 |
. . 3
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42 | 23, 41 | biimtrid 152 |
. 2
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43 | 2, 4, 6, 8, 19, 42, 10 | findcard2sd 6892 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iord 4367 df-on 4369 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-er 6535 df-en 6741 df-fin 6743 |
This theorem is referenced by: fimaxq 10807 |
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