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Mirrors > Home > MPE Home > Th. List > gt0ne0 | Structured version Visualization version GIF version |
Description: Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
gt0ne0 | ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 10380 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | ltne 10473 | . 2 ⊢ ((0 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
3 | 1, 2 | sylan 575 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 ≠ wne 2968 class class class wbr 4886 ℝcr 10271 0cc0 10272 < clt 10411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-addrcl 10333 ax-rnegex 10343 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 |
This theorem is referenced by: recgt0 11221 lemul1 11229 lediv1 11242 gt0div 11243 ge0div 11244 mulge0b 11247 ltdivmul 11252 ledivmul 11253 lt2mul2div 11255 lemuldiv 11257 ltdiv2 11263 ltrec1 11264 lerec2 11265 ledivdiv 11266 lediv2 11267 ltdiv23 11268 lediv23 11269 lediv12a 11270 recreclt 11276 nnrecl 11640 elnnz 11738 recnz 11804 rpne0 12155 divelunit 12631 resqrex 14398 sqrtgt0 14406 argregt0 24793 argimgt0 24795 logneg2 24798 logcnlem3 24827 atanlogsublem 25093 leopmul 29565 cdj1i 29864 lediv2aALT 32168 nndivlub 33040 knoppndvlem15 33099 knoppndvlem17 33101 sineq0ALT 40088 eenglngeehlnmlem1 43455 eenglngeehlnmlem2 43456 |
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