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Mirrors > Home > ILE Home > Th. List > gt0ne0d | GIF version |
Description: Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
gt0ne0d.1 | ⊢ (𝜑 → 0 < 𝐴) |
Ref | Expression |
---|---|
gt0ne0d | ⊢ (𝜑 → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7971 | . 2 ⊢ 0 ∈ ℝ | |
2 | gt0ne0d.1 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
3 | ltne 8056 | . 2 ⊢ ((0 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
4 | 1, 2, 3 | sylancr 414 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 ≠ wne 2357 class class class wbr 4015 ℝcr 7824 0cc0 7825 < clt 8006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1re 7919 ax-addrcl 7922 ax-rnegex 7934 ax-pre-ltirr 7937 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-pnf 8008 df-mnf 8009 df-ltxr 8011 |
This theorem is referenced by: sup3exmid 8928 modqval 10338 modqvalr 10339 modqcl 10340 flqpmodeq 10341 modq0 10343 modqge0 10346 modqlt 10347 modqdiffl 10349 modqdifz 10350 modqvalp1 10357 modqid 10363 modqcyc 10373 modqadd1 10375 modqmuladd 10380 modqmuladdnn0 10382 modqmul1 10391 modqdi 10406 modqsubdir 10407 ennnfonelemp1 12421 |
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