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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 8977 | . 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 2160 class class class wbr 4018 0cc0 7840 # cap 8567 ℕcn 8948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-inn 8949 |
This theorem is referenced by: qtri3or 10272 qbtwnrelemcalc 10285 intfracq 10350 flqdiv 10351 modqmulnn 10372 facndiv 10750 bcn0 10766 bcn1 10769 bcm1k 10771 bcp1n 10772 bcp1nk 10773 bcval5 10774 bcpasc 10777 permnn 10782 divcnv 11536 trireciplem 11539 trirecip 11540 expcnvap0 11541 geo2sum 11553 geo2lim 11555 cvgratnnlemfm 11568 cvgratnnlemrate 11569 mertenslemi1 11574 eftabs 11695 efcllemp 11697 ege2le3 11710 efcj 11712 efaddlem 11713 eftlub 11729 eirraplem 11815 dvdsflip 11888 dvdsgcdidd 12026 mulgcd 12048 gcddiv 12051 sqgcd 12061 lcmgcdlem 12108 qredeu 12128 prmind2 12151 isprm5lem 12172 divgcdodd 12174 sqrt2irrlem 12192 oddpwdclemxy 12200 oddpwdclemodd 12203 oddpwdclemdc 12204 sqrt2irraplemnn 12210 sqrt2irrap 12211 qmuldeneqnum 12226 divnumden 12227 numdensq 12233 hashdvds 12252 phiprmpw 12253 pythagtriplem19 12313 pcprendvds2 12322 pcpremul 12324 pceulem 12325 pceu 12326 pcdiv 12333 pcqmul 12334 pcid 12355 pc2dvds 12361 dvdsprmpweqle 12368 pcaddlem 12370 pcadd 12371 oddprmdvds 12385 pockthlem 12387 4sqlem5 12413 mul4sqlem 12424 4sqlem12 12433 4sqlem15 12436 4sqlem16 12437 4sqlem17 12438 logbgcd1irraplemexp 14838 logbgcd1irraplemap 14839 lgseisenlem2 14904 m1lgs 14905 2sqlem3 14917 2sqlem8 14923 cvgcmp2nlemabs 15234 redcwlpolemeq1 15256 |
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