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Mirrors > Home > ILE Home > Th. List > elnnz1 | GIF version |
Description: Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
elnnz1 | ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 9073 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | nnge1 8743 | . . 3 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
4 | 0lt1 7889 | . . . . 5 ⊢ 0 < 1 | |
5 | zre 9058 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | 0re 7766 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 1re 7765 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | ltletr 7853 | . . . . . . 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 ∈ ℤ → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) |
11 | 4, 10 | mpani 426 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 ≤ 𝑁 → 0 < 𝑁)) |
12 | 11 | imdistani 441 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 1 ≤ 𝑁) → (𝑁 ∈ ℤ ∧ 0 < 𝑁)) |
13 | elnnz 9064 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | |
14 | 12, 13 | sylibr 133 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | 3, 14 | impbii 125 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 0cc0 7620 1c1 7621 < clt 7800 ≤ cle 7801 ℕcn 8720 ℤcz 9054 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-z 9055 |
This theorem is referenced by: nnzrab 9078 znnnlt1 9102 eluz2b2 9397 elfznn 9834 flqge1nn 10067 resqrexlemdecn 10784 cvgratz 11301 oddennn 11905 |
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