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Mirrors > Home > MPE Home > Th. List > gt0ne0i | Structured version Visualization version 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 10357 | . 2 ⊢ 0 ∈ ℝ | |
2 | lt2.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
3 | 1, 2 | ltnei 10479 | 1 ⊢ (0 < 𝐴 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ≠ wne 2998 class class class wbr 4872 ℝcr 10250 0cc0 10251 < clt 10390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-resscn 10308 ax-1cn 10309 ax-addrcl 10312 ax-rnegex 10322 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-ltxr 10395 |
This theorem is referenced by: gt0ne0ii 10887 ledivp1i 11278 ltdivp1i 11279 |
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