![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10724 | . . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | |
2 | breq2 5034 | . . . . 5 ⊢ (𝐵 = 𝐴 → (𝐴 < 𝐵 ↔ 𝐴 < 𝐴)) | |
3 | 2 | notbid 321 | . . . 4 ⊢ (𝐵 = 𝐴 → (¬ 𝐴 < 𝐵 ↔ ¬ 𝐴 < 𝐴)) |
4 | 1, 3 | syl5ibrcom 250 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 = 𝐴 → ¬ 𝐴 < 𝐵)) |
5 | 4 | necon2ad 3002 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → 𝐵 ≠ 𝐴)) |
6 | 5 | imp 410 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ℝcr 10525 < clt 10664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: ltlen 10730 gtneii 10741 ltnei 10753 gtned 10764 gt0ne0 11094 lt0ne0 11095 gt0ne0d 11193 coprm 16045 phibndlem 16097 cshwshashlem1 16421 chfacffsupp 21461 chfacfscmul0 21463 chfacfscmulgsum 21465 chfacfpmmul0 21467 chfacfpmmulgsum 21469 sineq0 25116 logbgt0b 25379 axlowdimlem16 26751 frgrogt3nreg 28182 staddi 30029 stadd3i 30031 knoppndvlem12 33975 knoppndvlem14 33977 tan2h 35049 poimirlem24 35081 ftc1cnnc 35129 fdc 35183 60gcd7e1 39293 sineq0ALT 41643 sqrtnegnre 43864 requad01 44139 rrx2plord2 45136 eenglngeehlnmlem1 45151 |
Copyright terms: Public domain | W3C validator |