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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 8867 | . 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 2128 class class class wbr 3967 0cc0 7734 # cap 8460 ℕcn 8838 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-inn 8839 |
This theorem is referenced by: qtri3or 10151 qbtwnrelemcalc 10164 intfracq 10228 flqdiv 10229 modqmulnn 10250 facndiv 10624 bcn0 10640 bcn1 10643 bcm1k 10645 bcp1n 10646 bcp1nk 10647 bcval5 10648 bcpasc 10651 permnn 10656 divcnv 11405 trireciplem 11408 trirecip 11409 expcnvap0 11410 geo2sum 11422 geo2lim 11424 cvgratnnlemfm 11437 cvgratnnlemrate 11438 mertenslemi1 11443 eftabs 11564 efcllemp 11566 ege2le3 11579 efcj 11581 efaddlem 11582 eftlub 11598 eirraplem 11684 dvdsflip 11755 dvdsgcdidd 11893 mulgcd 11915 gcddiv 11918 sqgcd 11928 lcmgcdlem 11969 qredeu 11989 prmind2 12012 divgcdodd 12033 sqrt2irrlem 12051 oddpwdclemxy 12059 oddpwdclemodd 12062 oddpwdclemdc 12063 sqrt2irraplemnn 12069 sqrt2irrap 12070 qmuldeneqnum 12085 divnumden 12086 numdensq 12092 hashdvds 12111 phiprmpw 12112 pythagtriplem19 12172 pcprendvds2 12181 pcpremul 12183 pceulem 12184 pceu 12185 logbgcd1irraplemexp 13356 logbgcd1irraplemap 13357 cvgcmp2nlemabs 13674 redcwlpolemeq1 13696 |
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