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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 11264 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
| 2 | ltne 11358 | . 2 ⊢ ((0 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
| 3 | 1, 2 | sylan 580 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ℝcr 11154 0cc0 11155 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: recgt0 12113 lemul1 12119 lediv1 12133 gt0div 12134 ge0div 12135 mulge0b 12138 ltdivmul 12143 ledivmul 12144 lt2mul2div 12146 lemuldiv 12148 ltdiv2 12154 ltrec1 12155 lerec2 12156 ledivdiv 12157 lediv2 12158 ltdiv23 12159 lediv23 12160 lediv12a 12161 recreclt 12167 nnrecl 12524 elnnz 12623 recnz 12693 rpne0 13051 divelunit 13534 resqrex 15289 sqrtgt0 15297 argregt0 26652 argimgt0 26654 logneg2 26657 logcnlem3 26686 atanlogsublem 26958 leopmul 32153 cdj1i 32452 lediv2aALT 35682 nndivlub 36459 knoppndvlem15 36527 knoppndvlem17 36529 sineq0ALT 44957 eenglngeehlnmlem1 48658 eenglngeehlnmlem2 48659 |
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