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Mirrors > Home > ILE Home > Th. List > nnap0d | GIF version |
Description: A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnap0d | ⊢ (𝜑 → 𝐴 # 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnap0 8749 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 # 0) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐴 # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3929 0cc0 7620 # cap 8343 ℕcn 8720 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-inn 8721 |
This theorem is referenced by: qtri3or 10020 qbtwnrelemcalc 10033 intfracq 10093 flqdiv 10094 modqmulnn 10115 facndiv 10485 bcn0 10501 bcn1 10504 bcm1k 10506 bcp1n 10507 bcp1nk 10508 bcval5 10509 bcpasc 10512 permnn 10517 divcnv 11266 trireciplem 11269 trirecip 11270 expcnvap0 11271 geo2sum 11283 geo2lim 11285 cvgratnnlemfm 11298 cvgratnnlemrate 11299 mertenslemi1 11304 eftabs 11362 efcllemp 11364 ege2le3 11377 efcj 11379 efaddlem 11380 eftlub 11396 eirraplem 11483 dvdsflip 11549 dvdsgcdidd 11682 mulgcd 11704 gcddiv 11707 sqgcd 11717 lcmgcdlem 11758 qredeu 11778 prmind2 11801 divgcdodd 11821 sqrt2irrlem 11839 oddpwdclemxy 11847 oddpwdclemodd 11850 oddpwdclemdc 11851 sqrt2irraplemnn 11857 sqrt2irrap 11858 qmuldeneqnum 11873 divnumden 11874 numdensq 11880 hashdvds 11897 phiprmpw 11898 cvgcmp2nlemabs 13227 |
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