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| Mirrors > Home > MPE Home > Th. List > ltne | Structured version Visualization version GIF version | ||
| Description: 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
| Ref | Expression |
|---|---|
| ltne | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltnr 11356 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
| 2 | breq2 5147 | . . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐴)) | |
| 3 | 2 | notbid 318 | . . . 4 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴)) |
| 4 | 1, 3 | syl5ibrcom 247 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵)) |
| 5 | 4 | necon2ad 2955 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
| 6 | 5 | imp 406 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ℝcr 11154 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: ltlen 11362 gtneii 11373 ltnei 11385 gtned 11396 gt0ne0 11728 lt0ne0 11729 gt0ne0d 11827 coprm 16748 phibndlem 16807 cshwshashlem1 17133 chfacffsupp 22862 chfacfscmul0 22864 chfacfscmulgsum 22866 chfacfpmmul0 22868 chfacfpmmulgsum 22870 sineq0 26566 logbgt0b 26836 axlowdimlem16 28972 frgrogt3nreg 30416 staddi 32265 stadd3i 32267 knoppndvlem12 36524 knoppndvlem14 36526 tan2h 37619 poimirlem24 37651 ftc1cnnc 37699 fdc 37752 60gcd7e1 42006 sineq0ALT 44957 sqrtnegnre 47319 requad01 47608 rrx2plord2 48643 eenglngeehlnmlem1 48658 |
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