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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 9593 |
. 2
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2 | renemnf 7838 |
. . . . 5
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3 | 2 | neneqd 2330 |
. . . 4
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4 | mnflt 9599 |
. . . . 5
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5 | notnot 619 |
. . . . 5
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6 | 4, 5 | syl 14 |
. . . 4
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7 | 3, 6 | 2falsed 692 |
. . 3
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8 | pnfnemnf 7844 |
. . . . . 6
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9 | neeq1 2322 |
. . . . . 6
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10 | 8, 9 | mpbiri 167 |
. . . . 5
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11 | 10 | neneqd 2330 |
. . . 4
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12 | mnfltpnf 9601 |
. . . . . . 7
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13 | breq2 3941 |
. . . . . . 7
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14 | 12, 13 | mpbiri 167 |
. . . . . 6
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15 | 14 | necon3bi 2359 |
. . . . 5
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16 | 15 | necon2bi 2364 |
. . . 4
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17 | 11, 16 | 2falsed 692 |
. . 3
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18 | id 19 |
. . . 4
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19 | mnfxr 7846 |
. . . . . 6
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20 | xrltnr 9596 |
. . . . . 6
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21 | 19, 20 | ax-mp 5 |
. . . . 5
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22 | breq2 3941 |
. . . . 5
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23 | 21, 22 | mtbiri 665 |
. . . 4
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24 | 18, 23 | 2thd 174 |
. . 3
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25 | 7, 17, 24 | 3jaoi 1282 |
. 2
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26 | 1, 25 | sylbi 120 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 |
This theorem is referenced by: nmnfgt 9631 ge0nemnf 9637 xleaddadd 9700 |
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