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Mirrors > Home > MPE Home > Th. List > eluznn0 | Structured version Visualization version GIF version |
Description: Membership in a nonnegative upper set of integers implies membership in ℕ0. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
eluznn0 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12871 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 1 | uztrn2 12848 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ‘cfv 6543 0cc0 11116 ℕ0cn0 12479 ℤ≥cuz 12829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 |
This theorem is referenced by: elfz2nn0 13599 uzsubfz0 13616 leexp2r 14146 fi1uzind 14465 swrdlen2 14617 swrdfv2 14618 pfxccatpfx2 14694 geoserg 15819 geolim2 15824 geomulcvg 15829 mertenslem1 15837 mertenslem2 15838 mertens 15839 efcllem 16028 eftlcl 16057 reeftlcl 16058 eftlub 16059 efsep 16060 ruclem9 16188 smuval2 16430 smupvallem 16431 algfx 16524 eucalgcvga 16530 pcfaclem 16838 prmunb 16854 vdwlem7 16927 vdwlem10 16930 ramtlecl 16940 cpnord 25785 plyco0 26044 radcnvlem1 26264 abelthlem5 26287 abelthlem7 26290 log2tlbnd 26791 ftalem4 26921 ftalem5 26922 bcmono 27123 sseqp1 33858 subfaclim 34643 knoppndvlem6 35857 geomcau 37091 incssnn0 41912 jm2.27c 42209 iunrelexpuztr 42933 radcnvrat 43536 binomcxplemnn0 43571 stoweidlem7 45182 dignnld 47451 |
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