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Mirrors > Home > ILE Home > Th. List > xltnegi | Unicode version |
Description: Forward direction of xltneg 8519. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xltnegi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 8466 |
. . 3
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2 | elxr 8466 |
. . . . . 6
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3 | ltneg 7252 |
. . . . . . . . 9
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4 | rexneg 8513 |
. . . . . . . . . 10
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5 | rexneg 8513 |
. . . . . . . . . 10
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6 | 4, 5 | breqan12rd 3771 |
. . . . . . . . 9
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7 | 3, 6 | bitr4d 180 |
. . . . . . . 8
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8 | 7 | biimpd 132 |
. . . . . . 7
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9 | xnegeq 8510 |
. . . . . . . . . . 11
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10 | xnegpnf 8511 |
. . . . . . . . . . 11
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11 | 9, 10 | syl6eq 2085 |
. . . . . . . . . 10
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12 | 11 | adantl 262 |
. . . . . . . . 9
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13 | renegcl 7068 |
. . . . . . . . . . . 12
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14 | 5, 13 | eqeltrd 2111 |
. . . . . . . . . . 11
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15 | mnflt 8474 |
. . . . . . . . . . 11
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16 | 14, 15 | syl 14 |
. . . . . . . . . 10
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17 | 16 | adantr 261 |
. . . . . . . . 9
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18 | 12, 17 | eqbrtrd 3775 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 18 | a1d 22 |
. . . . . . 7
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20 | simpr 103 |
. . . . . . . . 9
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21 | 20 | breq2d 3767 |
. . . . . . . 8
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22 | rexr 6868 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | nltmnf 8479 |
. . . . . . . . . . 11
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24 | 22, 23 | syl 14 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | adantr 261 |
. . . . . . . . 9
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26 | 25 | pm2.21d 549 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 21, 26 | sylbid 139 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 8, 19, 27 | 3jaodan 1200 |
. . . . . 6
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29 | 2, 28 | sylan2b 271 |
. . . . 5
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30 | 29 | expimpd 345 |
. . . 4
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31 | simpl 102 |
. . . . . . 7
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32 | 31 | breq1d 3765 |
. . . . . 6
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33 | pnfnlt 8478 |
. . . . . . . 8
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34 | 33 | adantl 262 |
. . . . . . 7
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35 | 34 | pm2.21d 549 |
. . . . . 6
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36 | 32, 35 | sylbid 139 |
. . . . 5
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37 | 36 | expimpd 345 |
. . . 4
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38 | breq1 3758 |
. . . . . 6
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39 | 38 | anbi2d 437 |
. . . . 5
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40 | renegcl 7068 |
. . . . . . . . . . 11
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41 | 4, 40 | eqeltrd 2111 |
. . . . . . . . . 10
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42 | 41 | adantr 261 |
. . . . . . . . 9
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43 | ltpnf 8472 |
. . . . . . . . 9
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44 | 42, 43 | syl 14 |
. . . . . . . 8
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45 | 11 | adantr 261 |
. . . . . . . . 9
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46 | mnfltpnf 8476 |
. . . . . . . . 9
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47 | 45, 46 | syl6eqbr 3792 |
. . . . . . . 8
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48 | breq2 3759 |
. . . . . . . . . 10
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49 | mnfxr 8464 |
. . . . . . . . . . . 12
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50 | nltmnf 8479 |
. . . . . . . . . . . 12
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51 | 49, 50 | ax-mp 7 |
. . . . . . . . . . 11
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52 | 51 | pm2.21i 574 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 48, 52 | syl6bi 152 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 53 | imp 115 |
. . . . . . . 8
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55 | 44, 47, 54 | 3jaoian 1199 |
. . . . . . 7
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56 | 2, 55 | sylanb 268 |
. . . . . 6
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57 | xnegeq 8510 |
. . . . . . . 8
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58 | xnegmnf 8512 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() | |
59 | 57, 58 | syl6eq 2085 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 59 | breq2d 3767 |
. . . . . 6
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61 | 56, 60 | syl5ibr 145 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 39, 61 | sylbid 139 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
63 | 30, 37, 62 | 3jaoi 1197 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | 1, 63 | sylbi 114 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
65 | 64 | 3impib 1101 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-if 3326 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-sub 6981 df-neg 6982 df-xneg 8459 |
This theorem is referenced by: xltneg 8519 |
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