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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 11900 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ ℕ0) |
3 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | nn0ge0 11910 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
5 | elfz2nn0 12986 | . 2 ⊢ (0 ∈ (0...𝑁) ↔ (0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑁)) | |
6 | 2, 3, 4, 5 | syl3anbrc 1335 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 0cc0 10525 ≤ cle 10664 ℕ0cn0 11885 ...cfz 12880 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 |
This theorem is referenced by: fz0sn0fz1 13012 bcn0 13658 pfxmpt 14028 pfxfv 14032 pfxswrd 14056 swrdpfx 14057 pfxpfx 14058 pfxccatpfx1 14086 pfxccatpfx2 14087 pfxco 14188 chfacfscmulgsum 21396 chfacfpmmulgsum 21400 cayhamlem1 21402 wlkepvtx 27369 pthdadjvtx 27438 spthdep 27442 spthonepeq 27460 crctcsh 27529 wwlknllvtx 27551 wpthswwlks2on 27667 erclwwlknref 27775 0wlkonlem1 27824 upgr3v3e3cycl 27886 upgr4cycl4dv4e 27891 eupth2eucrct 27923 konigsbergiedgw 27954 konigsberglem1 27958 konigsberglem2 27959 konigsberglem3 27960 konigsberglem4 27961 cycpmco2f1 30693 circlemethhgt 31813 f1resfz0f1d 32258 pthhashvtx 32271 poimirlem5 34778 poimirlem20 34793 poimirlem22 34795 poimirlem28 34801 poimirlem32 34805 iccpartigtl 43460 iccpartlt 43461 iccpartgel 43466 iccpartrn 43467 iccelpart 43470 iccpartiun 43471 iccpartdisj 43474 |
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