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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 11761 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1 | uztrn2 11743 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2030 ‘cfv 5926 1c1 9975 ℕcn 11058 ℤ≥cuz 11725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-z 11416 df-uz 11726 |
This theorem is referenced by: elfzo1 12557 expmulnbnd 13036 bcval5 13145 isercolllem1 14439 isercoll 14442 o1fsum 14589 climcndslem1 14625 climcndslem2 14626 climcnds 14627 mertenslem2 14661 rpnnen2lem6 14992 rpnnen2lem7 14993 rpnnen2lem9 14995 rpnnen2lem11 14997 pcmpt2 15644 pcmptdvds 15645 prmreclem4 15670 prmreclem5 15671 prmreclem6 15672 vdwnnlem2 15747 2expltfac 15846 setsstructOLD 15946 1stcelcls 21312 lmnn 23107 cmetcaulem 23132 causs 23142 caubl 23152 caublcls 23153 ovolunlem1a 23310 volsuplem 23369 uniioombllem3 23399 mbfi1fseqlem6 23532 aaliou3lem2 24143 birthdaylem2 24724 lgamgulmlem4 24803 lgamcvg2 24826 chtub 24982 bclbnd 25050 bposlem3 25056 bposlem4 25057 bposlem5 25058 bposlem6 25059 lgsdilem2 25103 chebbnd1lem1 25203 chebbnd1lem2 25204 chebbnd1lem3 25205 dchrisumlema 25222 dchrisumlem2 25224 dchrisumlem3 25225 dchrisum0lem1b 25249 dchrisum0lem1 25250 pntrsumbnd2 25301 pntpbnd1 25320 pntpbnd2 25321 pntlemh 25333 pntlemq 25335 pntlemr 25336 pntlemj 25337 pntlemf 25339 minvecolem3 27860 minvecolem4 27864 h2hcau 27964 h2hlm 27965 chscllem2 28625 sinccvglem 31692 lmclim2 33684 geomcau 33685 heibor1lem 33738 rrncmslem 33761 divcnvg 40177 stoweidlem7 40542 stirlinglem12 40620 fourierdlem103 40744 fourierdlem104 40745 |
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