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Mirrors > Home > ILE Home > Th. List > ltne | Unicode version |
Description: 'Less than' implies not equal. See also ltap 7731 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
ltne |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7188 | . . . 4 | |
2 | breq2 3789 | . . . . 5 | |
3 | 2 | notbid 624 | . . . 4 |
4 | 1, 3 | syl5ibrcom 155 | . . 3 |
5 | 4 | necon2ad 2302 | . 2 |
6 | 5 | imp 122 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wceq 1284 wcel 1433 wne 2245 class class class wbr 3785 cr 6980 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 ax-pre-ltirr 7088 |
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-ltxr 7158 |
This theorem is referenced by: gtneii 7206 ltnei 7214 gtned 7223 gt0ne0 7531 lt0ne0 7532 gt0ne0d 7613 nngt1ne1 8073 zdceq 8423 qdceq 9256 coprm 10523 |
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