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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 9565 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1 | eleq2i 2244 | 1 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 ‘cfv 5218 1c1 7814 ℕcn 8921 ℤ≥cuz 9530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-z 9256 df-uz 9531 |
This theorem is referenced by: eluzge3nn 9574 uznnssnn 9579 elnndc 9614 uzsubsubfz1 10050 elfz1end 10057 fznn 10091 fzo1fzo0n0 10185 elfzonlteqm1 10212 rebtwn2z 10257 nnsinds 10445 exp3vallem 10523 exp1 10528 expp1 10529 facp1 10712 faclbnd 10723 bcn1 10740 resqrexlemf1 11019 resqrexlemfp1 11020 summodclem3 11390 summodclem2a 11391 fsum3 11397 fsumcl2lem 11408 fsumadd 11416 sumsnf 11419 fsummulc2 11458 trireciplem 11510 geo2lim 11526 geoisum1 11529 geoisum1c 11530 cvgratnnlemnexp 11534 cvgratz 11542 prodmodclem3 11585 prodmodclem2a 11586 fprodseq 11593 fprodmul 11601 prodsnf 11602 fprodfac 11625 dvdsfac 11868 gcdsupex 11960 gcdsupcl 11961 prmind2 12122 eulerthlemrprm 12231 eulerthlema 12232 pcmpt 12343 prmunb 12362 nninfdclemp1 12453 structfn 12483 mulg1 12995 mulgnndir 13017 lgsval2lem 14496 lgsdir 14521 lgsdilem2 14522 lgsdi 14523 lgsne0 14524 2sqlem10 14557 cvgcmp2nlemabs 14865 trilpolemisumle 14871 nconstwlpolem0 14896 |
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