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Mirrors > Home > ILE Home > Th. List > nltmnf | Unicode version |
Description: No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
nltmnf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfnre 7161 | . . . . . . 7 | |
2 | 1 | neli 2341 | . . . . . 6 |
3 | 2 | intnan 871 | . . . . 5 |
4 | 3 | intnanr 872 | . . . 4 |
5 | pnfnemnf 8851 | . . . . . 6 | |
6 | 5 | nesymi 2291 | . . . . 5 |
7 | 6 | intnan 871 | . . . 4 |
8 | 4, 7 | pm3.2ni 759 | . . 3 |
9 | 6 | intnan 871 | . . . 4 |
10 | 2 | intnan 871 | . . . 4 |
11 | 9, 10 | pm3.2ni 759 | . . 3 |
12 | 8, 11 | pm3.2ni 759 | . 2 |
13 | mnfxr 8848 | . . 3 | |
14 | ltxr 8849 | . . 3 | |
15 | 13, 14 | mpan2 415 | . 2 |
16 | 12, 15 | mtbiri 632 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 wceq 1284 wcel 1433 class class class wbr 3785 cr 6980 cltrr 6985 cpnf 7150 cmnf 7151 cxr 7152 clt 7153 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 |
This theorem is referenced by: mnfle 8867 xrltnsym 8868 xrlttr 8870 xrltso 8871 xltnegi 8902 qbtwnxr 9266 |
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