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Mirrors > Home > MPE Home > Th. List > rprene0d | Structured version Visualization version GIF version |
Description: A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rprene0d | ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
2 | 1 | rpred 12155 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 1 | rpne0d 12160 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) |
4 | 2, 3 | jca 509 | 1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ≠ wne 2998 ℝcr 10250 0cc0 10251 ℝ+crp 12111 |
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 df-rp 12112 |
This theorem is referenced by: circum 32111 |
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