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Mirrors > Home > ILE Home > Th. List > ltne | Unicode version |
Description: 'Less than' implies not equal. See also ltap 8106 which is the same but for apartness. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
ltne |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7560 |
. . . 4
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2 | breq2 3849 |
. . . . 5
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3 | 2 | notbid 627 |
. . . 4
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4 | 1, 3 | syl5ibrcom 155 |
. . 3
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5 | 4 | necon2ad 2312 |
. 2
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6 | 5 | imp 122 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-pre-ltirr 7455 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-xp 4444 df-pnf 7522 df-mnf 7523 df-ltxr 7525 |
This theorem is referenced by: gtneii 7578 ltnei 7586 gtned 7595 gt0ne0 7903 lt0ne0 7904 gt0ne0d 7988 nngt1ne1 8455 zdceq 8820 qdceq 9654 coprm 11397 phibndlem 11466 |
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