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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 10632 | . 2 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | ltne 10725 | . 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 396 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ℝcr 10524 0cc0 10525 < clt 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-addrcl 10586 ax-rnegex 10596 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 |
This theorem is referenced by: recgt0 11474 lemul1 11480 lediv1 11493 gt0div 11494 ge0div 11495 mulge0b 11498 ltdivmul 11503 ledivmul 11504 lt2mul2div 11506 lemuldiv 11508 ltdiv2 11514 ltrec1 11515 lerec2 11516 ledivdiv 11517 lediv2 11518 ltdiv23 11519 lediv23 11520 lediv12a 11521 recreclt 11527 nnrecl 11883 elnnz 11979 recnz 12045 rpne0 12393 divelunit 12868 resqrex 14598 sqrtgt0 14606 argregt0 25120 argimgt0 25122 logneg2 25125 logcnlem3 25154 atanlogsublem 25420 leopmul 29838 cdj1i 30137 lediv2aALT 32817 nndivlub 33703 knoppndvlem15 33762 knoppndvlem17 33764 sineq0ALT 41148 eenglngeehlnmlem1 44652 eenglngeehlnmlem2 44653 |
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