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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 9683 | . 2 | |
2 | renemnf 7926 | . . . . 5 | |
3 | 2 | neneqd 2348 | . . . 4 |
4 | mnflt 9690 | . . . . 5 | |
5 | notnot 619 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 3, 6 | 2falsed 692 | . . 3 |
8 | pnfnemnf 7932 | . . . . . 6 | |
9 | neeq1 2340 | . . . . . 6 | |
10 | 8, 9 | mpbiri 167 | . . . . 5 |
11 | 10 | neneqd 2348 | . . . 4 |
12 | mnfltpnf 9692 | . . . . . . 7 | |
13 | breq2 3969 | . . . . . . 7 | |
14 | 12, 13 | mpbiri 167 | . . . . . 6 |
15 | 14 | necon3bi 2377 | . . . . 5 |
16 | 15 | necon2bi 2382 | . . . 4 |
17 | 11, 16 | 2falsed 692 | . . 3 |
18 | id 19 | . . . 4 | |
19 | mnfxr 7934 | . . . . . 6 | |
20 | xrltnr 9686 | . . . . . 6 | |
21 | 19, 20 | ax-mp 5 | . . . . 5 |
22 | breq2 3969 | . . . . 5 | |
23 | 21, 22 | mtbiri 665 | . . . 4 |
24 | 18, 23 | 2thd 174 | . . 3 |
25 | 7, 17, 24 | 3jaoi 1285 | . 2 |
26 | 1, 25 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 w3o 962 wceq 1335 wcel 2128 wne 2327 class class class wbr 3965 cr 7731 cpnf 7909 cmnf 7910 cxr 7911 clt 7912 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 |
This theorem is referenced by: nmnfgt 9722 ge0nemnf 9728 xleaddadd 9791 |
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