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Mirrors > Home > ILE Home > Th. List > elnnuz | GIF version |
Description: A positive integer expressed as a member of an upper set of integers. (Contributed by NM, 6-Jun-2006.) |
Ref | Expression |
---|---|
elnnuz | ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 9479 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1 | eleq2i 2224 | 1 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2128 ‘cfv 5172 1c1 7735 ℕcn 8838 ℤ≥cuz 9444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-addcom 7834 ax-addass 7836 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-0id 7842 ax-rnegex 7843 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-ltadd 7850 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-mpt 4029 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-inn 8839 df-z 9173 df-uz 9445 |
This theorem is referenced by: eluzge3nn 9488 uznnssnn 9493 uzsubsubfz1 9956 elfz1end 9963 fznn 9997 fzo1fzo0n0 10091 elfzonlteqm1 10118 rebtwn2z 10163 nnsinds 10351 exp3vallem 10429 exp1 10434 expp1 10435 facp1 10615 faclbnd 10626 bcn1 10643 resqrexlemf1 10919 resqrexlemfp1 10920 summodclem3 11288 summodclem2a 11289 fsum3 11295 fsumcl2lem 11306 fsumadd 11314 sumsnf 11317 fsummulc2 11356 trireciplem 11408 geo2lim 11424 geoisum1 11427 geoisum1c 11428 cvgratnnlemnexp 11432 cvgratz 11440 prodmodclem3 11483 prodmodclem2a 11484 fprodseq 11491 fprodmul 11499 prodsnf 11500 fprodfac 11523 dvdsfac 11764 gcdsupex 11856 gcdsupcl 11857 prmind2 12012 eulerthlemrprm 12119 eulerthlema 12120 nninfdclemp1 12251 structfn 12279 cvgcmp2nlemabs 13674 trilpolemisumle 13680 nconstwlpolem0 13704 |
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