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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 12539 | . . 3 ⊢ 0 ∈ ℕ0 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ ℕ0) |
3 | id 22 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | nn0ge0 12549 | . 2 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
5 | elfz2nn0 13646 | . 2 ⊢ (0 ∈ (0...𝑁) ↔ (0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑁)) | |
6 | 2, 3, 4, 5 | syl3anbrc 1340 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5153 (class class class)co 7424 0cc0 11158 ≤ cle 11299 ℕ0cn0 12524 ...cfz 13538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 |
This theorem is referenced by: fz0sn0fz1 13672 bcn0 14327 pfxmpt 14686 pfxfv 14690 pfxswrd 14714 swrdpfx 14715 pfxpfx 14716 pfxccatpfx1 14744 pfxccatpfx2 14745 pfxco 14847 chfacfscmulgsum 22853 chfacfpmmulgsum 22857 cayhamlem1 22859 wlkepvtx 29597 pthdadjvtx 29667 spthdep 29671 spthonepeq 29689 crctcsh 29758 wwlknllvtx 29780 wpthswwlks2on 29895 erclwwlknref 30002 0wlkonlem1 30051 upgr3v3e3cycl 30113 upgr4cycl4dv4e 30118 eupth2eucrct 30150 konigsbergiedgw 30181 konigsberglem1 30185 konigsberglem2 30186 konigsberglem3 30187 konigsberglem4 30188 cycpmco2f1 33002 circlemethhgt 34489 f1resfz0f1d 34941 pthhashvtx 34955 poimirlem5 37326 poimirlem20 37341 poimirlem22 37343 poimirlem28 37349 poimirlem32 37353 prjspnfv01 42278 prjspner01 42279 prjspner1 42280 iccpartigtl 46995 iccpartlt 46996 iccpartgel 47001 iccpartrn 47002 iccelpart 47005 iccpartiun 47006 iccpartdisj 47009 |
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