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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 11224 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | ltne 11318 | . 2 ⊢ ((0 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
3 | 1, 2 | sylan 579 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ℝcr 11115 0cc0 11116 < clt 11255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-addrcl 11177 ax-rnegex 11187 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-ltxr 11260 |
This theorem is referenced by: recgt0 12067 lemul1 12073 lediv1 12086 gt0div 12087 ge0div 12088 mulge0b 12091 ltdivmul 12096 ledivmul 12097 lt2mul2div 12099 lemuldiv 12101 ltdiv2 12107 ltrec1 12108 lerec2 12109 ledivdiv 12110 lediv2 12111 ltdiv23 12112 lediv23 12113 lediv12a 12114 recreclt 12120 nnrecl 12477 elnnz 12575 recnz 12644 rpne0 12997 divelunit 13478 resqrex 15204 sqrtgt0 15212 argregt0 26458 argimgt0 26460 logneg2 26463 logcnlem3 26492 atanlogsublem 26761 leopmul 31821 cdj1i 32120 lediv2aALT 35127 nndivlub 35809 knoppndvlem15 35868 knoppndvlem17 35870 sineq0ALT 44163 eenglngeehlnmlem1 47587 eenglngeehlnmlem2 47588 |
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