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Mirrors > Home > ILE Home > Th. List > ngtmnft | Unicode version |
Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
ngtmnft |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9775 |
. 2
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2 | renemnf 8005 |
. . . . 5
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3 | 2 | neneqd 2368 |
. . . 4
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4 | mnflt 9782 |
. . . . 5
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5 | notnot 629 |
. . . . 5
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6 | 4, 5 | syl 14 |
. . . 4
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7 | 3, 6 | 2falsed 702 |
. . 3
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8 | pnfnemnf 8011 |
. . . . . 6
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9 | neeq1 2360 |
. . . . . 6
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10 | 8, 9 | mpbiri 168 |
. . . . 5
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11 | 10 | neneqd 2368 |
. . . 4
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12 | mnfltpnf 9784 |
. . . . . . 7
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13 | breq2 4007 |
. . . . . . 7
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14 | 12, 13 | mpbiri 168 |
. . . . . 6
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15 | 14 | necon3bi 2397 |
. . . . 5
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16 | 15 | necon2bi 2402 |
. . . 4
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17 | 11, 16 | 2falsed 702 |
. . 3
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18 | id 19 |
. . . 4
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19 | mnfxr 8013 |
. . . . . 6
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20 | xrltnr 9778 |
. . . . . 6
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21 | 19, 20 | ax-mp 5 |
. . . . 5
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22 | breq2 4007 |
. . . . 5
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23 | 21, 22 | mtbiri 675 |
. . . 4
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24 | 18, 23 | 2thd 175 |
. . 3
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25 | 7, 17, 24 | 3jaoi 1303 |
. 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 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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltirr 7922 |
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-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 |
This theorem is referenced by: nmnfgt 9817 ge0nemnf 9823 xleaddadd 9886 |
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