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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 8696 | . 2 | |
2 | renemnf 7074 | . . . . 5 | |
3 | 2 | neneqd 2226 | . . . 4 |
4 | mnflt 8704 | . . . . 5 | |
5 | notnot 559 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 3, 6 | 2falsed 618 | . . 3 |
8 | pnfnemnf 8697 | . . . . . 6 | |
9 | neeq1 2218 | . . . . . 6 | |
10 | 8, 9 | mpbiri 157 | . . . . 5 |
11 | 10 | neneqd 2226 | . . . 4 |
12 | mnfltpnf 8706 | . . . . . . 7 | |
13 | breq2 3768 | . . . . . . 7 | |
14 | 12, 13 | mpbiri 157 | . . . . . 6 |
15 | 14 | necon3bi 2255 | . . . . 5 |
16 | 15 | necon2bi 2260 | . . . 4 |
17 | 11, 16 | 2falsed 618 | . . 3 |
18 | id 19 | . . . 4 | |
19 | mnfxr 8694 | . . . . . 6 | |
20 | xrltnr 8701 | . . . . . 6 | |
21 | 19, 20 | ax-mp 7 | . . . . 5 |
22 | breq2 3768 | . . . . 5 | |
23 | 21, 22 | mtbiri 600 | . . . 4 |
24 | 18, 23 | 2thd 164 | . . 3 |
25 | 7, 17, 24 | 3jaoi 1198 | . 2 |
26 | 1, 25 | sylbi 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 98 w3o 884 wceq 1243 wcel 1393 wne 2204 class class class wbr 3764 cr 6888 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 |
This theorem is referenced by: ge0nemnf 8737 |
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