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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 7790 | . 2 ⊢ 0 ∈ ℝ | |
2 | gt0ne0d.1 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
3 | ltne 7873 | . 2 ⊢ ((0 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0) | |
4 | 1, 2, 3 | sylancr 411 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ≠ wne 2309 class class class wbr 3937 ℝcr 7643 0cc0 7644 < clt 7824 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-rnegex 7753 ax-pre-ltirr 7756 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-ltxr 7829 |
This theorem is referenced by: sup3exmid 8739 modqval 10128 modqvalr 10129 modqcl 10130 flqpmodeq 10131 modq0 10133 modqge0 10136 modqlt 10137 modqdiffl 10139 modqdifz 10140 modqvalp1 10147 modqid 10153 modqcyc 10163 modqadd1 10165 modqmuladd 10170 modqmuladdnn0 10172 modqmul1 10181 modqdi 10196 modqsubdir 10197 ennnfonelemp1 11955 |
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