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Mirrors > Home > MPE Home > Th. List > nn0fz0 | Structured version Visualization version GIF version |
Description: A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.) |
Ref | Expression |
---|---|
nn0fz0 | ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
2 | nn0re 11628 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
3 | 2 | leidd 10918 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ≤ 𝑁) |
4 | fznn0 12726 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ (0...𝑁) ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))) | |
5 | 1, 3, 4 | mpbir2and 706 | . 2 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0...𝑁)) |
6 | elfz3nn0 12728 | . 2 ⊢ (𝑁 ∈ (0...𝑁) → 𝑁 ∈ ℕ0) | |
7 | 5, 6 | impbii 201 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2166 class class class wbr 4873 (class class class)co 6905 0cc0 10252 ≤ cle 10392 ℕ0cn0 11618 ...cfz 12619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 |
This theorem is referenced by: swrdidOLD 13715 swrdrlen 13724 swrd0len0OLD 13725 pfxid 13763 pfxccat1 13781 swrd0swrdidOLD 13791 pfxpfxid 13792 pfxcctswrd 13793 wrdcctswrdOLD 13794 swrdccatwrdOLD 13800 pfxccatin12 13831 swrdccatin12OLD 13832 pfxccatid 13844 cshwlen 13920 cshwidxmod 13924 fallfacfac 15148 cayhamlem1 21041 cpmadugsumlemF 21051 wlkepvtx 26957 wlkp1lem7 26980 wlkp1lem8 26981 spthdep 27036 crctcshwlkn0lem6 27114 crctcsh 27123 wwlknllvtx 27145 wwlksnred 27202 wwlksnredOLD 27203 wpthswwlks2on 27290 clwlksfclwwlk1hashOLD 27436 konigsbergiedgw 27627 konigsberglem1 27631 konigsberglem2 27632 konigsberglem3 27633 dlwwlknondlwlknonf1olem1 27764 dlwwlknonclwlknonf1olem1OLD 27765 iwrdsplit 30994 iwrdsplitOLD 30995 fibp1 31009 poimirlem10 33963 poimirlem17 33970 poimirlem23 33976 poimirlem26 33979 poimirlem27 33980 iccpartiltu 42246 iccpartlt 42248 iccpartleu 42252 iccpartrn 42254 iccelpart 42257 iccpartiun 42258 iccpartdisj 42261 |
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