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Mirrors > Home > ILE Home > Th. List > lt0ne0d | GIF version |
Description: Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
lt0ne0d.1 | ⊢ (𝜑 → 𝐴 < 0) |
Ref | Expression |
---|---|
lt0ne0d | ⊢ (𝜑 → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt0ne0d.1 | . 2 ⊢ (𝜑 → 𝐴 < 0) | |
2 | 0re 7389 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | 2 | ltnri 7478 | . . . 4 ⊢ ¬ 0 < 0 |
4 | breq1 3814 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 < 0 ↔ 0 < 0)) | |
5 | 3, 4 | mtbiri 633 | . . 3 ⊢ (𝐴 = 0 → ¬ 𝐴 < 0) |
6 | 5 | necon2ai 2303 | . 2 ⊢ (𝐴 < 0 → 𝐴 ≠ 0) |
7 | 1, 6 | syl 14 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ≠ wne 2249 class class class wbr 3811 0cc0 7251 < clt 7423 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7337 ax-resscn 7338 ax-1re 7340 ax-addrcl 7343 ax-rnegex 7355 ax-pre-ltirr 7358 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-xp 4405 df-pnf 7425 df-mnf 7426 df-ltxr 7428 |
This theorem is referenced by: divalglemeuneg 10701 |
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