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Mirrors > Home > MPE Home > Th. List > 0elfz | Structured version Visualization version GIF version |
Description: 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.) |
Ref | Expression |
---|---|
0elfz | ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 11664 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ ℕ0) |
3 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | nn0ge0 11674 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
5 | elfz2nn0 12754 | . 2 ⊢ (0 ∈ (0...𝑁) ↔ (0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑁)) | |
6 | 2, 3, 4, 5 | syl3anbrc 1400 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 4888 (class class class)co 6924 0cc0 10274 ≤ cle 10414 ℕ0cn0 11647 ...cfz 12648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 |
This theorem is referenced by: fz0sn0fz1 12780 bcn0 13421 swrd0fOLD 13750 swrdidOLD 13751 swrd0fvOLD 13764 pfxmpt 13793 pfxfv 13797 swrd0swrdOLD 13821 pfxswrd 13822 swrdswrd0OLD 13823 swrdpfx 13824 swrd0swrd0OLD 13825 pfxpfx 13826 wrdcctswrdOLD 13830 swrdccatwrdOLD 13836 wrdeqs1catOLD 13846 swrdccatin12lem2OLD 13865 pfxccatpfx1 13873 pfxccatpfx2 13874 swrdccat3aOLD 13876 pfxco 13995 chfacfscmulgsum 21083 chfacfpmmulgsum 21087 cayhamlem1 21089 wlkepvtx 27024 pthdadjvtx 27099 spthdep 27103 spthonepeq 27121 crctcsh 27190 wwlknllvtx 27212 wpthswwlks2on 27358 erclwwlknref 27484 0wlkonlem1 27538 upgr3v3e3cycl 27600 upgr4cycl4dv4e 27605 eupth2eucrct 27638 konigsbergiedgw 27671 konigsberglem1 27675 konigsberglem2 27676 konigsberglem3 27677 konigsberglem4 27678 circlemethhgt 31331 poimirlem5 34049 poimirlem20 34064 poimirlem22 34066 poimirlem28 34072 poimirlem32 34076 iccpartigtl 42405 iccpartlt 42406 iccpartgel 42411 iccpartrn 42412 iccelpart 42415 iccpartiun 42416 iccpartdisj 42419 |
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