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Mirrors > Home > ILE Home > Th. List > xltnegi | Unicode version |
Description: Forward direction of xltneg 9823. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xltnegi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9763 |
. . 3
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2 | elxr 9763 |
. . . . . 6
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3 | ltneg 8409 |
. . . . . . . . 9
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4 | rexneg 9817 |
. . . . . . . . . 10
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5 | rexneg 9817 |
. . . . . . . . . 10
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6 | 4, 5 | breqan12rd 4017 |
. . . . . . . . 9
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7 | 3, 6 | bitr4d 191 |
. . . . . . . 8
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8 | 7 | biimpd 144 |
. . . . . . 7
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9 | xnegeq 9814 |
. . . . . . . . . . 11
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10 | xnegpnf 9815 |
. . . . . . . . . . 11
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11 | 9, 10 | eqtrdi 2226 |
. . . . . . . . . 10
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12 | 11 | adantl 277 |
. . . . . . . . 9
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13 | renegcl 8208 |
. . . . . . . . . . . 12
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14 | 5, 13 | eqeltrd 2254 |
. . . . . . . . . . 11
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15 | mnflt 9770 |
. . . . . . . . . . 11
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16 | 14, 15 | syl 14 |
. . . . . . . . . 10
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17 | 16 | adantr 276 |
. . . . . . . . 9
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18 | 12, 17 | eqbrtrd 4022 |
. . . . . . . 8
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19 | 18 | a1d 22 |
. . . . . . 7
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20 | simpr 110 |
. . . . . . . . 9
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21 | 20 | breq2d 4012 |
. . . . . . . 8
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22 | rexr 7993 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | nltmnf 9775 |
. . . . . . . . . . 11
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24 | 22, 23 | syl 14 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | adantr 276 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | pm2.21d 619 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 21, 26 | sylbid 150 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 8, 19, 27 | 3jaodan 1306 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 2, 28 | sylan2b 287 |
. . . . 5
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30 | 29 | expimpd 363 |
. . . 4
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31 | simpl 109 |
. . . . . . 7
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32 | 31 | breq1d 4010 |
. . . . . 6
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33 | pnfnlt 9774 |
. . . . . . . 8
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34 | 33 | adantl 277 |
. . . . . . 7
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35 | 34 | pm2.21d 619 |
. . . . . 6
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36 | 32, 35 | sylbid 150 |
. . . . 5
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37 | 36 | expimpd 363 |
. . . 4
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38 | breq1 4003 |
. . . . . 6
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39 | 38 | anbi2d 464 |
. . . . 5
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40 | renegcl 8208 |
. . . . . . . . . . 11
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41 | 4, 40 | eqeltrd 2254 |
. . . . . . . . . 10
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42 | 41 | adantr 276 |
. . . . . . . . 9
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43 | ltpnf 9767 |
. . . . . . . . 9
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44 | 42, 43 | syl 14 |
. . . . . . . 8
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45 | 11 | adantr 276 |
. . . . . . . . 9
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46 | mnfltpnf 9772 |
. . . . . . . . 9
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47 | 45, 46 | eqbrtrdi 4039 |
. . . . . . . 8
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48 | breq2 4004 |
. . . . . . . . . 10
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49 | mnfxr 8004 |
. . . . . . . . . . . 12
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50 | nltmnf 9775 |
. . . . . . . . . . . 12
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51 | 49, 50 | ax-mp 5 |
. . . . . . . . . . 11
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52 | 51 | pm2.21i 646 |
. . . . . . . . . 10
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53 | 48, 52 | syl6bi 163 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
54 | 53 | imp 124 |
. . . . . . . 8
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55 | 44, 47, 54 | 3jaoian 1305 |
. . . . . . 7
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56 | 2, 55 | sylanb 284 |
. . . . . 6
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57 | xnegeq 9814 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
58 | xnegmnf 9816 |
. . . . . . . 8
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59 | 57, 58 | eqtrdi 2226 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 59 | breq2d 4012 |
. . . . . 6
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61 | 56, 60 | syl5ibr 156 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 39, 61 | sylbid 150 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
63 | 30, 37, 62 | 3jaoi 1303 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | 1, 63 | sylbi 121 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
65 | 64 | 3impib 1201 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 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-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-sub 8120 df-neg 8121 df-xneg 9759 |
This theorem is referenced by: xltneg 9823 |
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