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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9563 | . 2 | |
2 | renemnf 7814 | . . . . 5 | |
3 | 2 | neneqd 2329 | . . . 4 |
4 | mnflt 9569 | . . . . 5 | |
5 | notnot 618 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 3, 6 | 2falsed 691 | . . 3 |
8 | pnfnemnf 7820 | . . . . . 6 | |
9 | neeq1 2321 | . . . . . 6 | |
10 | 8, 9 | mpbiri 167 | . . . . 5 |
11 | 10 | neneqd 2329 | . . . 4 |
12 | mnfltpnf 9571 | . . . . . . 7 | |
13 | breq2 3933 | . . . . . . 7 | |
14 | 12, 13 | mpbiri 167 | . . . . . 6 |
15 | 14 | necon3bi 2358 | . . . . 5 |
16 | 15 | necon2bi 2363 | . . . 4 |
17 | 11, 16 | 2falsed 691 | . . 3 |
18 | id 19 | . . . 4 | |
19 | mnfxr 7822 | . . . . . 6 | |
20 | xrltnr 9566 | . . . . . 6 | |
21 | 19, 20 | ax-mp 5 | . . . . 5 |
22 | breq2 3933 | . . . . 5 | |
23 | 21, 22 | mtbiri 664 | . . . 4 |
24 | 18, 23 | 2thd 174 | . . 3 |
25 | 7, 17, 24 | 3jaoi 1281 | . 2 |
26 | 1, 25 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 w3o 961 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 cr 7619 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 |
This theorem is referenced by: nmnfgt 9601 ge0nemnf 9607 xleaddadd 9670 |
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