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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 8370 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | nnge1 8062 | . . 3 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
3 | 1, 2 | jca 300 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
4 | 0lt1 7236 | . . . . 5 ⊢ 0 < 1 | |
5 | zre 8355 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | 0re 7119 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
7 | 1re 7118 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
8 | ltletr 7200 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) | |
9 | 6, 7, 8 | mp3an12 1258 | . . . . . 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 420 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 ≤ 𝑁 → 0 < 𝑁)) |
12 | 11 | imdistani 433 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 1 ≤ 𝑁) → (𝑁 ∈ ℤ ∧ 0 < 𝑁)) |
13 | elnnz 8361 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | |
14 | 12, 13 | sylibr 132 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 ≤ 𝑁) → 𝑁 ∈ ℕ) |
15 | 3, 14 | impbii 124 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 class class class wbr 3785 ℝcr 6980 0cc0 6981 1c1 6982 < clt 7153 ≤ cle 7154 ℕcn 8039 ℤcz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-z 8352 |
This theorem is referenced by: nnzrab 8375 znnnlt1 8399 eluz2b2 8690 elfznn 9073 flqge1nn 9296 resqrexlemdecn 9898 |
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