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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 9153 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1 | eleq2i 2161 | 1 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1445 ‘cfv 5049 1c1 7448 ℕcn 8520 ℤ≥cuz 9118 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-z 8849 df-uz 9119 |
This theorem is referenced by: eluzge3nn 9159 uznnssnn 9164 uzsubsubfz1 9611 elfz1end 9618 fznn 9652 fzo1fzo0n0 9743 elfzonlteqm1 9770 rebtwn2z 9815 nnsinds 9998 exp3vallem 10087 exp1 10092 expp1 10093 facp1 10269 faclbnd 10280 bcn1 10297 resqrexlemf1 10572 resqrexlemfp1 10573 summodclem3 10939 summodclem2a 10940 fsum3 10946 fsumcl2lem 10957 fsumadd 10965 sumsnf 10968 fsummulc2 11007 trireciplem 11059 geo2lim 11075 geoisum1 11078 geoisum1c 11079 cvgratnnlemnexp 11083 cvgratz 11091 dvdsfac 11304 gcdsupex 11392 gcdsupcl 11393 prmind2 11545 structfn 11678 |
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