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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 9522 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1 | eleq2i 2237 | 1 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2141 ‘cfv 5198 1c1 7775 ℕcn 8878 ℤ≥cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-z 9213 df-uz 9488 |
This theorem is referenced by: eluzge3nn 9531 uznnssnn 9536 elnndc 9571 uzsubsubfz1 10004 elfz1end 10011 fznn 10045 fzo1fzo0n0 10139 elfzonlteqm1 10166 rebtwn2z 10211 nnsinds 10399 exp3vallem 10477 exp1 10482 expp1 10483 facp1 10664 faclbnd 10675 bcn1 10692 resqrexlemf1 10972 resqrexlemfp1 10973 summodclem3 11343 summodclem2a 11344 fsum3 11350 fsumcl2lem 11361 fsumadd 11369 sumsnf 11372 fsummulc2 11411 trireciplem 11463 geo2lim 11479 geoisum1 11482 geoisum1c 11483 cvgratnnlemnexp 11487 cvgratz 11495 prodmodclem3 11538 prodmodclem2a 11539 fprodseq 11546 fprodmul 11554 prodsnf 11555 fprodfac 11578 dvdsfac 11820 gcdsupex 11912 gcdsupcl 11913 prmind2 12074 eulerthlemrprm 12183 eulerthlema 12184 pcmpt 12295 prmunb 12314 nninfdclemp1 12405 structfn 12435 lgsval2lem 13705 lgsdir 13730 lgsdilem2 13731 lgsdi 13732 lgsne0 13733 2sqlem10 13755 cvgcmp2nlemabs 14064 trilpolemisumle 14070 nconstwlpolem0 14094 |
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