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Mirrors > Home > MPE Home > Th. List > nn0ge0 | Structured version Visualization version GIF version |
Description: A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn0ge0 | ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11332 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nngt0 11087 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
3 | id 22 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 = 0) | |
4 | 3 | eqcomd 2657 | . . . 4 ⊢ (𝑁 = 0 → 0 = 𝑁) |
5 | 2, 4 | orim12i 537 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (0 < 𝑁 ∨ 0 = 𝑁)) |
6 | 1, 5 | sylbi 207 | . 2 ⊢ (𝑁 ∈ ℕ0 → (0 < 𝑁 ∨ 0 = 𝑁)) |
7 | 0re 10078 | . . 3 ⊢ 0 ∈ ℝ | |
8 | nn0re 11339 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
9 | leloe 10162 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) | |
10 | 7, 8, 9 | sylancr 696 | . 2 ⊢ (𝑁 ∈ ℕ0 → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) |
11 | 6, 10 | mpbird 247 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 382 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ℝcr 9973 0cc0 9974 < clt 10112 ≤ cle 10113 ℕcn 11058 ℕ0cn0 11330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 |
This theorem is referenced by: nn0nlt0 11357 nn0ge0i 11358 nn0le0eq0 11359 nn0p1gt0 11360 0mnnnnn0 11363 nn0addge1 11377 nn0addge2 11378 nn0negleid 11383 nn0ge0d 11392 nn0ge0div 11484 xnn0ge0 12005 nn0pnfge0OLD 12006 xnn0xadd0 12115 nn0rp0 12317 xnn0xrge0 12363 0elfz 12475 fz0fzelfz0 12484 fz0fzdiffz0 12487 fzctr 12490 difelfzle 12491 fzoun 12544 nn0p1elfzo 12550 elfzodifsumelfzo 12573 fvinim0ffz 12627 subfzo0 12630 adddivflid 12659 modmuladdnn0 12754 addmodid 12758 modifeq2int 12772 modfzo0difsn 12782 bernneq 13030 bernneq3 13032 faclbnd 13117 faclbnd6 13126 facubnd 13127 bcval5 13145 hashneq0 13193 fi1uzind 13317 brfi1indALT 13320 ccat0 13394 ccat2s1fvw 13460 repswswrd 13577 rprisefaccl 14798 dvdseq 15083 evennn02n 15121 nn0ehalf 15142 nn0oddm1d2 15148 bitsinv1 15211 smuval2 15251 gcdn0gt0 15286 nn0gcdid0 15289 absmulgcd 15313 algcvgblem 15337 algcvga 15339 lcmgcdnn 15371 lcmfun 15405 lcmfass 15406 nonsq 15514 hashgcdlem 15540 odzdvds 15547 pcfaclem 15649 coe1sclmul 19700 coe1sclmul2 19702 prmirredlem 19889 prmirred 19891 fvmptnn04ifb 20704 mdegle0 23882 plypf1 24013 dgrlt 24067 fta1 24108 taylfval 24158 eldmgm 24793 basellem3 24854 bcmono 25047 lgsdinn0 25115 dchrisumlem1 25223 dchrisumlem2 25224 wwlksnextwrd 26860 wwlksnextfun 26861 wwlksnextinj 26862 wwlksnextproplem2 26873 wwlksnextproplem3 26874 nn0sqeq1 29641 xrsmulgzz 29806 hashf2 30274 hasheuni 30275 reprinfz1 30828 faclimlem1 31755 rrntotbnd 33765 pell14qrgt0 37740 pell1qrgaplem 37754 monotoddzzfi 37824 jm2.17a 37844 jm2.22 37879 rmxdiophlem 37899 wallispilem3 40602 stirlinglem7 40615 elfz2z 41650 fz0addge0 41654 elfzlble 41655 2ffzoeq 41663 iccpartigtl 41684 sqrtpwpw2p 41775 flsqrt 41833 nn0e 41933 nn0sumltlt 42453 nn0eo 42647 fllog2 42687 dignn0fr 42720 dignnld 42722 dig1 42727 |
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