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Mirrors > Home > ILE Home > Th. List > elnnnn0c | GIF version |
Description: The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.) |
Ref | Expression |
---|---|
elnnnn0c | ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 8977 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | nnge1 8736 | . . 3 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) |
4 | 0lt1 7882 | . . . . 5 ⊢ 0 < 1 | |
5 | nn0re 8979 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 0re 7759 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 1re 7758 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | ltletr 7846 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) | |
9 | 6, 7, 8 | mp3an12 1305 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) |
10 | 5, 9 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) |
11 | 4, 10 | mpani 426 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ 𝑁 → 0 < 𝑁)) |
12 | 11 | imdistani 441 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁) → (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
13 | elnnnn0b 9014 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) | |
14 | 12, 13 | sylibr 133 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | 3, 14 | impbii 125 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 class class class wbr 3924 ℝcr 7612 0cc0 7613 1c1 7614 < clt 7793 ≤ cle 7794 ℕcn 8713 ℕ0cn0 8970 |
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-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 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-int 3767 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-inn 8714 df-n0 8971 |
This theorem is referenced by: nn0ge2m1nn 9030 nn0o1gt2 11591 |
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