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Mirrors > Home > MPE Home > Th. List > eluznn | Structured version Visualization version GIF version |
Description: Membership in a positive upper set of integers implies membership in ℕ. (Contributed by JJ, 1-Oct-2018.) |
Ref | Expression |
---|---|
eluznn | ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 11967 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1 | uztrn2 11948 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ‘cfv 6101 1c1 10225 ℕcn 11312 ℤ≥cuz 11930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-z 11667 df-uz 11931 |
This theorem is referenced by: elfzo1 12773 expmulnbnd 13250 bcval5 13358 isercolllem1 14736 isercoll 14739 o1fsum 14883 climcndslem1 14919 climcndslem2 14920 climcnds 14921 mertenslem2 14954 rpnnen2lem6 15284 rpnnen2lem7 15285 rpnnen2lem9 15287 rpnnen2lem11 15289 pcmpt2 15930 pcmptdvds 15931 prmreclem4 15956 prmreclem5 15957 prmreclem6 15958 vdwnnlem2 16033 2expltfac 16127 1stcelcls 21593 lmnn 23389 cmetcaulem 23414 causs 23424 caubl 23434 caublcls 23435 ovolunlem1a 23604 volsuplem 23663 uniioombllem3 23693 mbfi1fseqlem6 23828 aaliou3lem2 24439 birthdaylem2 25031 lgamgulmlem4 25110 lgamcvg2 25133 chtub 25289 bclbnd 25357 bposlem3 25363 bposlem4 25364 bposlem5 25365 bposlem6 25366 lgsdilem2 25410 chebbnd1lem1 25510 chebbnd1lem2 25511 chebbnd1lem3 25512 dchrisumlema 25529 dchrisumlem2 25531 dchrisumlem3 25532 dchrisum0lem1b 25556 dchrisum0lem1 25557 pntrsumbnd2 25608 pntpbnd1 25627 pntpbnd2 25628 pntlemh 25640 pntlemq 25642 pntlemr 25643 pntlemj 25644 pntlemf 25646 minvecolem3 28257 minvecolem4 28261 h2hcau 28361 h2hlm 28362 chscllem2 29022 sinccvglem 32081 lmclim2 34041 geomcau 34042 heibor1lem 34095 rrncmslem 34118 divcnvg 40603 stoweidlem7 40967 stirlinglem12 41045 fourierdlem103 41169 fourierdlem104 41170 |
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