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Mirrors > Home > ILE Home > Th. List > ge0p1rp | GIF version |
Description: A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
ge0p1rp | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2re 7891 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
2 | 1 | adantr 274 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ) |
3 | 0red 7760 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ∈ ℝ) | |
4 | simpl 108 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ) | |
5 | simpr 109 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ 𝐴) | |
6 | ltp1 8595 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) | |
7 | 6 | adantr 274 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 < (𝐴 + 1)) |
8 | 3, 4, 2, 5, 7 | lelttrd 7880 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 < (𝐴 + 1)) |
9 | elrp 9436 | . 2 ⊢ ((𝐴 + 1) ∈ ℝ+ ↔ ((𝐴 + 1) ∈ ℝ ∧ 0 < (𝐴 + 1))) | |
10 | 2, 8, 9 | sylanbrc 413 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℝcr 7612 0cc0 7613 1c1 7614 + caddc 7616 < clt 7793 ≤ cle 7794 ℝ+crp 9434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltwlin 7726 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-rp 9435 |
This theorem is referenced by: ge0p1rpd 9507 |
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