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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 9382 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 1 | rpne0d 9387 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) |
4 | 2, 3 | jca 302 | 1 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1463 ≠ wne 2282 ℝcr 7546 0cc0 7547 ℝ+crp 9343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1re 7639 ax-addrcl 7642 ax-rnegex 7654 ax-pre-ltirr 7657 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-pnf 7726 df-mnf 7727 df-ltxr 7729 df-rp 9344 |
This theorem is referenced by: (None) |
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