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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 8907 | . 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 2141 class class class wbr 3989 0cc0 7774 # cap 8500 ℕcn 8878 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-inn 8879 |
This theorem is referenced by: qtri3or 10199 qbtwnrelemcalc 10212 intfracq 10276 flqdiv 10277 modqmulnn 10298 facndiv 10673 bcn0 10689 bcn1 10692 bcm1k 10694 bcp1n 10695 bcp1nk 10696 bcval5 10697 bcpasc 10700 permnn 10705 divcnv 11460 trireciplem 11463 trirecip 11464 expcnvap0 11465 geo2sum 11477 geo2lim 11479 cvgratnnlemfm 11492 cvgratnnlemrate 11493 mertenslemi1 11498 eftabs 11619 efcllemp 11621 ege2le3 11634 efcj 11636 efaddlem 11637 eftlub 11653 eirraplem 11739 dvdsflip 11811 dvdsgcdidd 11949 mulgcd 11971 gcddiv 11974 sqgcd 11984 lcmgcdlem 12031 qredeu 12051 prmind2 12074 isprm5lem 12095 divgcdodd 12097 sqrt2irrlem 12115 oddpwdclemxy 12123 oddpwdclemodd 12126 oddpwdclemdc 12127 sqrt2irraplemnn 12133 sqrt2irrap 12134 qmuldeneqnum 12149 divnumden 12150 numdensq 12156 hashdvds 12175 phiprmpw 12176 pythagtriplem19 12236 pcprendvds2 12245 pcpremul 12247 pceulem 12248 pceu 12249 pcdiv 12256 pcqmul 12257 pcid 12277 pc2dvds 12283 dvdsprmpweqle 12290 pcaddlem 12292 pcadd 12293 oddprmdvds 12306 pockthlem 12308 4sqlem5 12334 mul4sqlem 12345 logbgcd1irraplemexp 13680 logbgcd1irraplemap 13681 2sqlem3 13747 2sqlem8 13753 cvgcmp2nlemabs 14064 redcwlpolemeq1 14086 |
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