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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 9212 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | nnge1 8971 | . . 3 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
3 | 1, 2 | jca 306 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) |
4 | 0lt1 8113 | . . . . 5 ⊢ 0 < 1 | |
5 | nn0re 9214 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 0re 7986 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 1re 7985 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | ltletr 8076 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) | |
9 | 6, 7, 8 | mp3an12 1338 | . . . . . 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 430 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ 𝑁 → 0 < 𝑁)) |
12 | 11 | imdistani 445 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁) → (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
13 | elnnnn0b 9249 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) | |
14 | 12, 13 | sylibr 134 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | 3, 14 | impbii 126 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 class class class wbr 4018 ℝcr 7839 0cc0 7840 1c1 7841 < clt 8021 ≤ cle 8022 ℕcn 8948 ℕ0cn0 9205 |
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 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
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-int 3860 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-iota 5196 df-fv 5243 df-ov 5898 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-inn 8949 df-n0 9206 |
This theorem is referenced by: nn0ge2m1nn 9265 nn0o1gt2 11941 pcelnn 12352 lgsabs1 14893 |
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