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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 10646 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | 2 | ltnri 10752 | . . . 4 ⊢ ¬ 0 < 0 |
4 | breq1 5072 | . . . 4 ⊢ (𝐴 = 0 → (𝐴 < 0 ↔ 0 < 0)) | |
5 | 3, 4 | mtbiri 329 | . . 3 ⊢ (𝐴 = 0 → ¬ 𝐴 < 0) |
6 | 5 | necon2ai 3048 | . 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 1536 ≠ wne 3019 class class class wbr 5069 0cc0 10540 < clt 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-addrcl 10601 ax-rnegex 10611 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 |
This theorem is referenced by: nnne0 11674 mul2lt0rlt0 12494 mbfmulc2lem 24251 coseq00topi 25091 argimlt0 25199 atantan 25504 bcm1n 30521 sgnmul 31804 sgnsub 31806 sgn0bi 31809 sgnmulsgn 31811 signsvfnn 31860 reclt0d 41664 requad1 43794 |
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