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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 7734 | . 2 ⊢ 0 ∈ ℝ | |
2 | gt0ne0d.1 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
3 | ltne 7817 | . 2 ⊢ ((0 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
4 | 1, 2, 3 | sylancr 410 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ≠ wne 2285 class class class wbr 3899 ℝcr 7587 0cc0 7588 < clt 7768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-rnegex 7697 ax-pre-ltirr 7700 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-pnf 7770 df-mnf 7771 df-ltxr 7773 |
This theorem is referenced by: sup3exmid 8683 modqval 10065 modqvalr 10066 modqcl 10067 flqpmodeq 10068 modq0 10070 modqge0 10073 modqlt 10074 modqdiffl 10076 modqdifz 10077 modqvalp1 10084 modqid 10090 modqcyc 10100 modqadd1 10102 modqmuladd 10107 modqmuladdnn0 10109 modqmul1 10118 modqdi 10133 modqsubdir 10134 ennnfonelemp1 11846 |
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