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Mirrors > Home > MPE Home > Th. List > elnnz1 | Structured version Visualization version 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 12595 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | nnge1 12256 | . . 3 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁) | |
3 | 1, 2 | jca 511 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
4 | 0lt1 11752 | . . . . 5 ⊢ 0 < 1 | |
5 | 0re 11232 | . . . . . 6 ⊢ 0 ∈ ℝ | |
6 | 1re 11230 | . . . . . 6 ⊢ 1 ∈ ℝ | |
7 | zre 12578 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
8 | ltletr 11322 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) | |
9 | 5, 6, 7, 8 | mp3an12i 1462 | . . . . 5 ⊢ (𝑁 ∈ ℤ → ((0 < 1 ∧ 1 ≤ 𝑁) → 0 < 𝑁)) |
10 | 4, 9 | mpani 695 | . . . 4 ⊢ (𝑁 ∈ ℤ → (1 ≤ 𝑁 → 0 < 𝑁)) |
11 | 10 | imdistani 568 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 1 ≤ 𝑁) → (𝑁 ∈ ℤ ∧ 0 < 𝑁)) |
12 | elnnz 12584 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | |
13 | 11, 12 | sylibr 233 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 1 ≤ 𝑁) → 𝑁 ∈ ℕ) |
14 | 3, 13 | impbii 208 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 class class class wbr 5142 ℝcr 11123 0cc0 11124 1c1 11125 < clt 11264 ≤ cle 11265 ℕcn 12228 ℤcz 12574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-z 12575 |
This theorem is referenced by: znnnlt1 12605 nnzrab 12606 eluz2b2 12921 elfznn 13548 elfz1b 13588 flge1nn 13804 gcdcllem3 16461 4sqlem11 16909 ovolunlem1a 25399 ovoliunlem1 25405 ppinncl 27080 bcmono 27184 zabsle1 27203 gausslemma2dlem1a 27272 gausslemma2dlem4 27276 axlowdimlem16 28742 nndiffz1 32525 tgoldbachgnn 34214 poimirlem7 37022 lcmineqlem13 41436 fz1eqin 42101 lzenom 42102 dirkertrigeqlem3 45401 |
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