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Mirrors > Home > ILE Home > Th. List > nn0z | GIF version |
Description: A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
nn0z | ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssz 9096 | . 2 ⊢ ℕ0 ⊆ ℤ | |
2 | 1 | sseli 3098 | 1 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ℕ0cn0 9001 ℤcz 9078 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: nn0negz 9112 nn0ltp1le 9140 nn0leltp1 9141 nn0ltlem1 9142 nn0sub 9144 nn0n0n1ge2b 9154 nn0lt10b 9155 nn0lt2 9156 nn0le2is012 9157 nn0lem1lt 9158 fnn0ind 9191 nn0pzuz 9409 nn01to3 9436 nn0ge2m1nnALT 9437 fz1n 9855 ige2m1fz 9921 elfz2nn0 9923 fznn0 9924 elfz0add 9931 fzctr 9941 difelfzle 9942 fzo1fzo0n0 9991 fzofzim 9996 elfzodifsumelfzo 10009 zpnn0elfzo 10015 fzossfzop1 10020 ubmelm1fzo 10034 adddivflid 10096 fldivnn0 10099 divfl0 10100 flqmulnn0 10103 fldivnn0le 10107 zmodidfzoimp 10158 modqmuladdnn0 10172 modifeq2int 10190 modfzo0difsn 10199 uzennn 10240 expdivap 10375 faclbnd3 10521 bccmpl 10532 bcnp1n 10537 bcn2 10542 bcp1m1 10543 modfsummodlemstep 11258 bcxmas 11290 geo2sum2 11316 mertenslemi1 11336 mertensabs 11338 esum 11405 efcvgfsum 11410 ege2le3 11414 eftlcl 11431 reeftlcl 11432 eftlub 11433 effsumlt 11435 eirraplem 11519 dvds1 11587 dvdsext 11589 addmodlteqALT 11593 oddnn02np1 11613 oddge22np1 11614 nn0ehalf 11636 nn0o1gt2 11638 nno 11639 nn0o 11640 nn0oddm1d2 11642 modremain 11662 gcdn0gt0 11702 nn0gcdid0 11705 bezoutlemmain 11722 nn0seqcvgd 11758 algcvgblem 11766 algcvga 11768 eucalgf 11772 prmndvdsfaclt 11870 nn0sqrtelqelz 11920 nonsq 11921 crth 11936 |
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