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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 11309 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | breq2 5153 | . . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐴)) | |
3 | 2 | notbid 318 | . . . 4 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴)) |
4 | 1, 3 | syl5ibrcom 246 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵)) |
5 | 4 | necon2ad 2956 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
6 | 5 | imp 408 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ℝcr 11109 < clt 11248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 |
This theorem is referenced by: ltlen 11315 gtneii 11326 ltnei 11338 gtned 11349 gt0ne0 11679 lt0ne0 11680 gt0ne0d 11778 coprm 16648 phibndlem 16703 cshwshashlem1 17029 chfacffsupp 22358 chfacfscmul0 22360 chfacfscmulgsum 22362 chfacfpmmul0 22364 chfacfpmmulgsum 22366 sineq0 26033 logbgt0b 26298 axlowdimlem16 28215 frgrogt3nreg 29650 staddi 31499 stadd3i 31501 knoppndvlem12 35399 knoppndvlem14 35401 tan2h 36480 poimirlem24 36512 ftc1cnnc 36560 fdc 36613 60gcd7e1 40870 sineq0ALT 43698 sqrtnegnre 46015 requad01 46289 rrx2plord2 47408 eenglngeehlnmlem1 47423 |
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