|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 12922 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
| 2 | 1 | uztrn2 12898 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ‘cfv 6560 1c1 11157 ℕcn 12267 ℤ≥cuz 12879 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-z 12616 df-uz 12880 | 
| This theorem is referenced by: elfzo1 13753 expmulnbnd 14275 bcval5 14358 isercolllem1 15702 isercoll 15705 o1fsum 15850 climcndslem1 15886 climcndslem2 15887 climcnds 15888 mertenslem2 15922 rpnnen2lem6 16256 rpnnen2lem7 16257 rpnnen2lem9 16259 rpnnen2lem11 16261 pcmpt2 16932 pcmptdvds 16933 prmreclem4 16958 prmreclem5 16959 prmreclem6 16960 vdwnnlem2 17035 2expltfac 17131 1stcelcls 23470 lmnn 25298 cmetcaulem 25323 causs 25333 caubl 25343 caublcls 25344 ovolunlem1a 25532 volsuplem 25591 uniioombllem3 25621 mbfi1fseqlem6 25756 aaliou3lem2 26386 birthdaylem2 26996 lgamgulmlem4 27076 lgamcvg2 27099 chtub 27257 bclbnd 27325 bposlem3 27331 bposlem4 27332 bposlem5 27333 bposlem6 27334 lgsdilem2 27378 chebbnd1lem1 27514 chebbnd1lem2 27515 chebbnd1lem3 27516 dchrisumlema 27533 dchrisumlem2 27535 dchrisumlem3 27536 dchrisum0lem1b 27560 dchrisum0lem1 27561 pntrsumbnd2 27612 pntpbnd1 27631 pntpbnd2 27632 pntlemh 27644 pntlemq 27646 pntlemr 27647 pntlemj 27648 pntlemf 27650 minvecolem3 30896 minvecolem4 30900 h2hcau 30999 h2hlm 31000 chscllem2 31658 sinccvglem 35678 lmclim2 37766 geomcau 37767 heibor1lem 37817 rrncmslem 37840 aks4d1p1 42078 fimgmcyc 42549 divcnvg 45647 stoweidlem7 46027 stirlinglem12 46105 fourierdlem103 46229 fourierdlem104 46230 | 
| Copyright terms: Public domain | W3C validator |