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Mirrors > Home > ILE Home > Th. List > xnegneg | Unicode version |
Description: Extended real version of negneg 8238. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegneg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9808 |
. 2
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2 | rexneg 9862 |
. . . . 5
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3 | xnegeq 9859 |
. . . . 5
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4 | 2, 3 | syl 14 |
. . . 4
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5 | renegcl 8249 |
. . . . 5
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6 | rexneg 9862 |
. . . . 5
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7 | 5, 6 | syl 14 |
. . . 4
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8 | recn 7975 |
. . . . 5
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9 | 8 | negnegd 8290 |
. . . 4
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10 | 4, 7, 9 | 3eqtrd 2226 |
. . 3
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11 | xnegmnf 9861 |
. . . 4
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12 | xnegeq 9859 |
. . . . . 6
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13 | xnegpnf 9860 |
. . . . . 6
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14 | 12, 13 | eqtrdi 2238 |
. . . . 5
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15 | xnegeq 9859 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | syl 14 |
. . . 4
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17 | id 19 |
. . . 4
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18 | 11, 16, 17 | 3eqtr4a 2248 |
. . 3
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19 | xnegeq 9859 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19, 11 | eqtrdi 2238 |
. . . . 5
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21 | xnegeq 9859 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 20, 21 | syl 14 |
. . . 4
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23 | id 19 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 13, 22, 23 | 3eqtr4a 2248 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 10, 18, 24 | 3jaoi 1314 |
. 2
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26 | 1, 25 | sylbi 121 |
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-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-sub 8161 df-neg 8162 df-xneg 9804 |
This theorem is referenced by: xneg11 9866 xltneg 9868 xnegdi 9900 xnpcan 9904 xrnegiso 11305 infxrnegsupex 11306 xrnegcon1d 11307 xrminmax 11308 xrmin1inf 11310 xrmin2inf 11311 xrltmininf 11313 xrlemininf 11314 xrminltinf 11315 xrminadd 11318 |
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