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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 11255 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | breq2 5110 | . . . . 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 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 2940 class class class wbr 5106 ℝcr 11055 < clt 11194 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 |
This theorem is referenced by: ltlen 11261 gtneii 11272 ltnei 11284 gtned 11295 gt0ne0 11625 lt0ne0 11626 gt0ne0d 11724 coprm 16592 phibndlem 16647 cshwshashlem1 16973 chfacffsupp 22221 chfacfscmul0 22223 chfacfscmulgsum 22225 chfacfpmmul0 22227 chfacfpmmulgsum 22229 sineq0 25896 logbgt0b 26159 axlowdimlem16 27948 frgrogt3nreg 29383 staddi 31230 stadd3i 31232 knoppndvlem12 35032 knoppndvlem14 35034 tan2h 36116 poimirlem24 36148 ftc1cnnc 36196 fdc 36250 60gcd7e1 40508 sineq0ALT 43307 sqrtnegnre 45625 requad01 45899 rrx2plord2 46894 eenglngeehlnmlem1 46909 |
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