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Mirrors > Home > ILE Home > Th. List > gt0ne0i | GIF version |
Description: Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.) |
Ref | Expression |
---|---|
lt2.1 | ⊢ 𝐴 ∈ ℝ |
Ref | Expression |
---|---|
gt0ne0i | ⊢ (0 < 𝐴 → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7878 | . 2 ⊢ 0 ∈ ℝ | |
2 | lt2.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
3 | 1, 2 | ltnei 7980 | 1 ⊢ (0 < 𝐴 → 𝐴 ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 ≠ wne 2327 class class class wbr 3965 ℝcr 7731 0cc0 7732 < clt 7912 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1re 7826 ax-addrcl 7829 ax-rnegex 7841 ax-pre-ltirr 7844 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-pnf 7914 df-mnf 7915 df-ltxr 7917 |
This theorem is referenced by: gt0ne0ii 8362 |
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