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Mirrors > Home > ILE Home > Th. List > ltne | Unicode version |
Description: 'Less than' implies not equal. See also ltap 8395 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 7841 | . . . 4 | |
2 | breq2 3933 | . . . . 5 | |
3 | 2 | notbid 656 | . . . 4 |
4 | 1, 3 | syl5ibrcom 156 | . . 3 |
5 | 4 | necon2ad 2365 | . 2 |
6 | 5 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 cr 7619 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-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-ltxr 7805 |
This theorem is referenced by: gtneii 7859 ltnei 7867 gtned 7876 gt0ne0 8189 lt0ne0 8190 gt0ne0d 8274 nngt1ne1 8755 zdceq 9126 qdceq 10024 coprm 11822 phibndlem 11892 |
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