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Mirrors > Home > ILE Home > Th. List > nnz | GIF version |
Description: A positive integer is an integer. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nnz | ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssz 8923 | . 2 ⊢ ℕ ⊆ ℤ | |
2 | 1 | sseli 3043 | 1 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1448 ℕcn 8578 ℤcz 8906 |
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 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-ltadd 7611 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-inn 8579 df-z 8907 |
This theorem is referenced by: elnnz1 8929 znegcl 8937 nnleltp1 8965 nnltp1le 8966 elz2 8974 nnlem1lt 8987 nnltlem1 8988 nnm1ge0 8989 prime 9002 nneo 9006 zeo 9008 btwnz 9022 indstr 9238 eluz2b2 9247 elnn1uz2 9251 qaddcl 9277 qreccl 9284 elfz1end 9676 fznatpl1 9697 fznn 9710 elfz1b 9711 elfzo0 9800 fzo1fzo0n0 9801 elfzo0z 9802 elfzo1 9808 ubmelm1fzo 9844 intfracq 9934 zmodcl 9958 zmodfz 9960 zmodfzo 9961 zmodid2 9966 zmodidfzo 9967 modfzo0difsn 10009 mulexpzap 10174 nnesq 10252 expnlbnd 10257 expnlbnd2 10258 facdiv 10325 faclbnd 10328 bc0k 10343 bcval5 10350 seq3coll 10426 caucvgrelemcau 10592 resqrexlemlo 10625 resqrexlemcalc3 10628 resqrexlemgt0 10632 absexpzap 10692 climuni 10901 fsum3 10995 arisum 11106 trireciplem 11108 expcnvap0 11110 geo2sum 11122 geo2lim 11124 0.999... 11129 geoihalfsum 11130 cvgratz 11140 dvdsval3 11292 nndivdvds 11294 modmulconst 11320 dvdsle 11337 dvdsssfz1 11345 fzm1ndvds 11349 dvdsfac 11353 oexpneg 11369 nnoddm1d2 11402 divalg2 11418 divalgmod 11419 modremain 11421 ndvdsadd 11423 nndvdslegcd 11449 divgcdz 11455 divgcdnn 11458 divgcdnnr 11459 modgcd 11474 gcddiv 11500 gcdmultiple 11501 gcdmultiplez 11502 gcdzeq 11503 gcdeq 11504 rpmulgcd 11507 rplpwr 11508 rppwr 11509 sqgcd 11510 dvdssqlem 11511 dvdssq 11512 eucalginv 11530 lcmgcdlem 11551 lcmgcdnn 11556 lcmass 11559 coprmgcdb 11562 qredeq 11570 qredeu 11571 cncongr1 11577 cncongr2 11578 1idssfct 11589 isprm2lem 11590 isprm3 11592 isprm4 11593 prmind2 11594 divgcdodd 11614 isprm6 11618 sqrt2irr 11633 pw2dvds 11636 sqrt2irraplemnn 11649 divnumden 11666 divdenle 11667 nn0gcdsq 11670 phivalfi 11680 phicl2 11682 phiprmpw 11690 hashgcdlem 11695 hashgcdeq 11696 oddennn 11697 evenennn 11698 unennn 11702 trilpolemcl 12814 |
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