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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 7638 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | 2 | ltnri 7727 | . . . 4 ⊢ ¬ 0 < 0 |
4 | breq1 3878 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 < 0 ↔ 0 < 0)) | |
5 | 3, 4 | mtbiri 641 | . . 3 ⊢ (𝐴 = 0 → ¬ 𝐴 < 0) |
6 | 5 | necon2ai 2321 | . 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 1299 ≠ wne 2267 class class class wbr 3875 0cc0 7500 < clt 7672 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 ax-rnegex 7604 ax-pre-ltirr 7607 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-xp 4483 df-pnf 7674 df-mnf 7675 df-ltxr 7677 |
This theorem is referenced by: divalglemeuneg 11415 |
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