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Mirrors > Home > MPE Home > Th. List > lt0ne0d | Structured version Visualization version 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 11157 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | 2 | ltnri 11264 | . . . 4 ⊢ ¬ 0 < 0 |
4 | breq1 5108 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 < 0 ↔ 0 < 0)) | |
5 | 3, 4 | mtbiri 326 | . . 3 ⊢ (𝐴 = 0 → ¬ 𝐴 < 0) |
6 | 5 | necon2ai 2973 | . 2 ⊢ (𝐴 < 0 → 𝐴 ≠ 0) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ≠ wne 2943 class class class wbr 5105 0cc0 11051 < clt 11189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-addrcl 11112 ax-rnegex 11122 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-ltxr 11194 |
This theorem is referenced by: nnne0 12187 mul2lt0rlt0 13017 mbfmulc2lem 25011 coseq00topi 25859 argimlt0 25968 atantan 26273 bcm1n 31698 sgnmul 33142 sgnsub 33144 sgn0bi 33147 sgnmulsgn 33149 signsvfnn 33198 sn-nnne0 40903 reclt0d 43611 requad1 45804 |
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