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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 11900 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnnz 11992 | . . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁)) | |
3 | eqcom 2828 | . . 3 ⊢ (𝑁 = 0 ↔ 0 = 𝑁) | |
4 | 2, 3 | orbi12i 911 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ↔ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) ∨ 0 = 𝑁)) |
5 | id 22 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
6 | 0z 11993 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
7 | eleq1 2900 | . . . . . . 7 ⊢ (0 = 𝑁 → (0 ∈ ℤ ↔ 𝑁 ∈ ℤ)) | |
8 | 6, 7 | mpbii 235 | . . . . . 6 ⊢ (0 = 𝑁 → 𝑁 ∈ ℤ) |
9 | 5, 8 | jaoi 853 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∨ 0 = 𝑁) → 𝑁 ∈ ℤ) |
10 | orc 863 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 ∈ ℤ ∨ 0 = 𝑁)) | |
11 | 9, 10 | impbii 211 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∨ 0 = 𝑁) ↔ 𝑁 ∈ ℤ) |
12 | 11 | anbi1i 625 | . . 3 ⊢ (((𝑁 ∈ ℤ ∨ 0 = 𝑁) ∧ (0 < 𝑁 ∨ 0 = 𝑁)) ↔ (𝑁 ∈ ℤ ∧ (0 < 𝑁 ∨ 0 = 𝑁))) |
13 | ordir 1003 | . . 3 ⊢ (((𝑁 ∈ ℤ ∧ 0 < 𝑁) ∨ 0 = 𝑁) ↔ ((𝑁 ∈ ℤ ∨ 0 = 𝑁) ∧ (0 < 𝑁 ∨ 0 = 𝑁))) | |
14 | 0re 10643 | . . . . 5 ⊢ 0 ∈ ℝ | |
15 | zre 11986 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
16 | leloe 10727 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) | |
17 | 14, 15, 16 | sylancr 589 | . . . 4 ⊢ (𝑁 ∈ ℤ → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) |
18 | 17 | pm5.32i 577 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ (0 < 𝑁 ∨ 0 = 𝑁))) |
19 | 12, 13, 18 | 3bitr4i 305 | . 2 ⊢ (((𝑁 ∈ ℤ ∧ 0 < 𝑁) ∨ 0 = 𝑁) ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
20 | 1, 4, 19 | 3bitri 299 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ℝcr 10536 0cc0 10537 < clt 10675 ≤ cle 10676 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 |
This theorem is referenced by: zle0orge1 11999 nn0zrab 12012 znn0sub 12030 nn0ind 12078 fnn0ind 12082 fznn0 13000 elfz0ubfz0 13012 elfz0fzfz0 13013 fz0fzelfz0 13014 elfzmlbp 13019 difelfzle 13021 difelfznle 13022 elfzo0z 13080 fzofzim 13085 ubmelm1fzo 13134 flge0nn0 13191 zmodcl 13260 modmuladdnn0 13284 modsumfzodifsn 13313 zsqcl2 13503 swrdnnn0nd 14018 swrdswrdlem 14066 swrdswrd 14067 swrdccatin2 14091 pfxccatin12lem2 14093 pfxccatin12lem3 14094 repswswrd 14146 cshwidxmod 14165 nn0abscl 14672 iseralt 15041 binomrisefac 15396 oexpneg 15694 oddnn02np1 15697 evennn02n 15699 nn0ehalf 15729 nn0oddm1d2 15736 divalglem2 15746 divalglem8 15751 divalglem10 15753 divalgb 15755 bitsinv1lem 15790 dfgcd2 15894 algcvga 15923 hashgcdlem 16125 iserodd 16172 pockthlem 16241 4sqlem14 16294 cshwshashlem2 16430 chfacfscmul0 21466 chfacfpmmul0 21470 taylfvallem1 24945 tayl0 24950 basellem3 25660 bcmono 25853 gausslemma2dlem0h 25939 2sqnn0 26014 crctcshwlkn0lem7 27594 crctcshwlkn0 27599 clwlkclwwlklem2a1 27770 clwlkclwwlklem2fv2 27774 clwlkclwwlklem2a 27776 wwlksubclwwlk 27837 0nn0m1nnn0 32351 knoppndvlem2 33852 irrapxlem1 39439 rmynn0 39574 rmyabs 39575 jm2.22 39612 jm2.23 39613 jm2.27a 39622 jm2.27c 39624 dvnprodlem1 42251 wallispilem4 42373 stirlinglem5 42383 elaa2lem 42538 etransclem3 42542 etransclem7 42546 etransclem10 42549 etransclem19 42558 etransclem20 42559 etransclem21 42560 etransclem22 42561 etransclem24 42563 etransclem27 42566 zm1nn 43522 eluzge0nn0 43532 elfz2z 43535 2elfz2melfz 43538 subsubelfzo0 43546 oexpnegALTV 43862 nn0oALTV 43881 nn0e 43882 nn0eo 44608 dig1 44688 |
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