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Mirrors > Home > ILE Home > Th. List > rprene0d | 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 9714 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 1 | rpne0d 9719 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) |
4 | 2, 3 | jca 306 | 1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 ≠ wne 2360 ℝcr 7828 0cc0 7829 ℝ+crp 9671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 ax-rnegex 7938 ax-pre-ltirr 7941 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4647 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-rp 9672 |
This theorem is referenced by: (None) |
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