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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 11888 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nngt0 11657 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
3 | id 22 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 = 0) | |
4 | 3 | eqcomd 2827 | . . . 4 ⊢ (𝑁 = 0 → 0 = 𝑁) |
5 | 2, 4 | orim12i 902 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (0 < 𝑁 ∨ 0 = 𝑁)) |
6 | 1, 5 | sylbi 218 | . 2 ⊢ (𝑁 ∈ ℕ0 → (0 < 𝑁 ∨ 0 = 𝑁)) |
7 | 0re 10632 | . . 3 ⊢ 0 ∈ ℝ | |
8 | nn0re 11895 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
9 | leloe 10716 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) | |
10 | 7, 8, 9 | sylancr 587 | . 2 ⊢ (𝑁 ∈ ℕ0 → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁))) |
11 | 6, 10 | mpbird 258 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ℝcr 10525 0cc0 10526 < clt 10664 ≤ cle 10665 ℕcn 11627 ℕ0cn0 11886 |
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 |
This theorem is referenced by: nn0nlt0 11912 nn0ge0i 11913 nn0le0eq0 11914 nn0p1gt0 11915 0mnnnnn0 11918 nn0addge1 11932 nn0addge2 11933 nn0negleid 11938 nn0ge0d 11947 nn0ge0div 12040 xnn0ge0 12518 xnn0xadd0 12630 nn0rp0 12833 xnn0xrge0 12881 0elfz 12994 fz0fzelfz0 13003 fz0fzdiffz0 13006 fzctr 13009 difelfzle 13010 fzoun 13064 nn0p1elfzo 13070 elfzodifsumelfzo 13093 fvinim0ffz 13146 subfzo0 13149 adddivflid 13178 modmuladdnn0 13273 addmodid 13277 modifeq2int 13291 modfzo0difsn 13301 nn0sq11 13487 bernneq 13580 bernneq3 13582 faclbnd 13640 faclbnd6 13649 facubnd 13650 bcval5 13668 hashneq0 13715 fi1uzind 13845 brfi1indALT 13848 ccat0 13919 ccat2s1fvw 13988 ccat2s1fvwOLD 13989 repswswrd 14136 nn0sqeq1 14626 rprisefaccl 15367 dvdseq 15654 evennn02n 15689 nn0ehalf 15719 nn0oddm1d2 15726 bitsinv1 15781 smuval2 15821 gcdn0gt0 15856 nn0gcdid0 15859 absmulgcd 15887 algcvgblem 15911 algcvga 15913 lcmgcdnn 15945 lcmfun 15979 lcmfass 15980 2mulprm 16027 nonsq 16089 hashgcdlem 16115 odzdvds 16122 pcfaclem 16224 coe1sclmul 20380 coe1sclmul2 20382 prmirredlem 20570 prmirred 20572 fvmptnn04ifb 21389 mdegle0 24600 plypf1 24731 dgrlt 24785 fta1 24826 taylfval 24876 logbgcd1irr 25299 eldmgm 25527 basellem3 25588 bcmono 25781 lgsdinn0 25849 2sq2 25937 2sqnn0 25942 2sqreulem1 25950 dchrisumlem1 25993 dchrisumlem2 25994 wwlksnextwrd 27603 wwlksnextfun 27604 wwlksnextinj 27605 wwlksnextproplem2 27617 wwlksnextproplem3 27618 wrdt2ind 30555 xrsmulgzz 30593 hashf2 31243 hasheuni 31244 reprinfz1 31793 0nn0m1nnn0 32249 faclimlem1 32873 rrntotbnd 34997 pell14qrgt0 39336 pell1qrgaplem 39350 monotoddzzfi 39419 jm2.17a 39437 jm2.22 39472 rmxdiophlem 39492 wallispilem3 42233 stirlinglem7 42246 elfz2z 43396 fz0addge0 43400 elfzlble 43401 2ffzoeq 43409 iccpartigtl 43430 sqrtpwpw2p 43547 flsqrt 43603 nn0e 43709 nn0sumltlt 44296 nn0eo 44486 fllog2 44526 dignn0fr 44559 dignnld 44561 dig1 44566 |
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