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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 7759 | . 2 ⊢ 0 ∈ ℝ | |
2 | gt0ne0d.1 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
3 | ltne 7842 | . 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 1480 ≠ wne 2306 class class class wbr 3924 ℝcr 7612 0cc0 7613 < clt 7793 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722 ax-pre-ltirr 7725 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: sup3exmid 8708 modqval 10090 modqvalr 10091 modqcl 10092 flqpmodeq 10093 modq0 10095 modqge0 10098 modqlt 10099 modqdiffl 10101 modqdifz 10102 modqvalp1 10109 modqid 10115 modqcyc 10125 modqadd1 10127 modqmuladd 10132 modqmuladdnn0 10134 modqmul1 10143 modqdi 10158 modqsubdir 10159 ennnfonelemp1 11908 |
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