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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 11906 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ ℕ0) |
3 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | nn0ge0 11916 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
5 | elfz2nn0 12992 | . 2 ⊢ (0 ∈ (0...𝑁) ↔ (0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑁)) | |
6 | 2, 3, 4, 5 | syl3anbrc 1339 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 0cc0 10531 ≤ cle 10670 ℕ0cn0 11891 ...cfz 12886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 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 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 |
This theorem is referenced by: fz0sn0fz1 13018 bcn0 13664 pfxmpt 14034 pfxfv 14038 pfxswrd 14062 swrdpfx 14063 pfxpfx 14064 pfxccatpfx1 14092 pfxccatpfx2 14093 pfxco 14194 chfacfscmulgsum 21462 chfacfpmmulgsum 21466 cayhamlem1 21468 wlkepvtx 27436 pthdadjvtx 27505 spthdep 27509 spthonepeq 27527 crctcsh 27596 wwlknllvtx 27618 wpthswwlks2on 27734 erclwwlknref 27842 0wlkonlem1 27891 upgr3v3e3cycl 27953 upgr4cycl4dv4e 27958 eupth2eucrct 27990 konigsbergiedgw 28021 konigsberglem1 28025 konigsberglem2 28026 konigsberglem3 28027 konigsberglem4 28028 cycpmco2f1 30761 circlemethhgt 31909 f1resfz0f1d 32356 pthhashvtx 32369 poimirlem5 34891 poimirlem20 34906 poimirlem22 34908 poimirlem28 34914 poimirlem32 34918 iccpartigtl 43577 iccpartlt 43578 iccpartgel 43583 iccpartrn 43584 iccelpart 43587 iccpartiun 43588 iccpartdisj 43591 |
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