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Mirrors > Home > MPE Home > Th. List > elnn0z | Structured version Visualization version GIF version |
Description: Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
elnn0z | ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11888 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnnz 11980 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | |
3 | eqcom 2828 | . . 3 ⊢ (𝑁 = 0 ↔ 0 = 𝑁) | |
4 | 2, 3 | orbi12i 908 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ↔ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) ∨ 0 = 𝑁)) |
5 | id 22 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
6 | 0z 11981 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | eleq1 2900 | . . . . . . 7 ⊢ (0 = 𝑁 → (0 ∈ ℤ ↔ 𝑁 ∈ ℤ)) | |
8 | 6, 7 | mpbii 234 | . . . . . 6 ⊢ (0 = 𝑁 → 𝑁 ∈ ℤ) |
9 | 5, 8 | jaoi 851 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∨ 0 = 𝑁) → 𝑁 ∈ ℤ) |
10 | orc 861 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 ∈ ℤ ∨ 0 = 𝑁)) | |
11 | 9, 10 | impbii 210 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∨ 0 = 𝑁) ↔ 𝑁 ∈ ℤ) |
12 | 11 | anbi1i 623 | . . 3 ⊢ (((𝑁 ∈ ℤ ∨ 0 = 𝑁) ∧ (0 < 𝑁 ∨ 0 = 𝑁)) ↔ (𝑁 ∈ ℤ ∧ (0 < 𝑁 ∨ 0 = 𝑁))) |
13 | ordir 1000 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 0 < 𝑁) ∨ 0 = 𝑁) ↔ ((𝑁 ∈ ℤ ∨ 0 = 𝑁) ∧ (0 < 𝑁 ∨ 0 = 𝑁))) | |
14 | 0re 10632 | . . . . 5 ⊢ 0 ∈ ℝ | |
15 | zre 11974 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
16 | leloe 10716 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) | |
17 | 14, 15, 16 | sylancr 587 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) |
18 | 17 | pm5.32i 575 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ (0 < 𝑁 ∨ 0 = 𝑁))) |
19 | 12, 13, 18 | 3bitr4i 304 | . 2 ⊢ (((𝑁 ∈ ℤ ∧ 0 < 𝑁) ∨ 0 = 𝑁) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
20 | 1, 4, 19 | 3bitri 298 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ℝcr 10525 0cc0 10526 < clt 10664 ≤ cle 10665 ℕcn 11627 ℕ0cn0 11886 ℤcz 11970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-z 11971 |
This theorem is referenced by: zle0orge1 11987 nn0zrab 12000 znn0sub 12018 nn0ind 12066 fnn0ind 12070 fznn0 12989 elfz0ubfz0 13001 elfz0fzfz0 13002 fz0fzelfz0 13003 elfzmlbp 13008 difelfzle 13010 difelfznle 13011 elfzo0z 13069 fzofzim 13074 ubmelm1fzo 13123 flge0nn0 13180 zmodcl 13249 modmuladdnn0 13273 modsumfzodifsn 13302 zsqcl2 13492 swrdnnn0nd 14008 swrdswrdlem 14056 swrdswrd 14057 swrdccatin2 14081 pfxccatin12lem2 14083 pfxccatin12lem3 14084 repswswrd 14136 cshwidxmod 14155 nn0abscl 14662 iseralt 15031 binomrisefac 15386 oexpneg 15684 oddnn02np1 15687 evennn02n 15689 nn0ehalf 15719 nn0oddm1d2 15726 divalglem2 15736 divalglem8 15741 divalglem10 15743 divalgb 15745 bitsinv1lem 15780 dfgcd2 15884 algcvga 15913 hashgcdlem 16115 iserodd 16162 pockthlem 16231 4sqlem14 16284 cshwshashlem2 16420 chfacfscmul0 21396 chfacfpmmul0 21400 taylfvallem1 24874 tayl0 24879 leibpilem1OLD 25446 basellem3 25588 bcmono 25781 gausslemma2dlem0h 25867 2sqnn0 25942 crctcshwlkn0lem7 27522 crctcshwlkn0 27527 clwlkclwwlklem2a1 27698 clwlkclwwlklem2fv2 27702 clwlkclwwlklem2a 27704 wwlksubclwwlk 27765 0nn0m1nnn0 32249 knoppndvlem2 33750 irrapxlem1 39299 rmynn0 39434 rmyabs 39435 jm2.22 39472 jm2.23 39473 jm2.27a 39482 jm2.27c 39484 dvnprodlem1 42111 wallispilem4 42234 stirlinglem5 42244 elaa2lem 42399 etransclem3 42403 etransclem7 42407 etransclem10 42410 etransclem19 42419 etransclem20 42420 etransclem21 42421 etransclem22 42422 etransclem24 42424 etransclem27 42427 zm1nn 43383 eluzge0nn0 43393 elfz2z 43396 2elfz2melfz 43399 subsubelfzo0 43407 oexpnegALTV 43689 nn0oALTV 43708 nn0e 43709 nn0eo 44486 dig1 44566 |
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