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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 9008 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | nnge1 8767 | . . 3 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁)) |
4 | 0lt1 7913 | . . . . 5 ⊢ 0 < 1 | |
5 | nn0re 9010 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 0re 7790 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 1re 7789 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | ltletr 7877 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) | |
9 | 6, 7, 8 | mp3an12 1306 | . . . . . 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 427 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ 𝑁 → 0 < 𝑁)) |
12 | 11 | imdistani 442 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁) → (𝑁 ∈ ℕ0 ∧ 0 < 𝑁)) |
13 | elnnnn0b 9045 | . . 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 1481 class class class wbr 3937 ℝcr 7643 0cc0 7644 1c1 7645 < clt 7824 ≤ cle 7825 ℕcn 8744 ℕ0cn0 9001 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-inn 8745 df-n0 9002 |
This theorem is referenced by: nn0ge2m1nn 9061 nn0o1gt2 11638 |
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