![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nnap0d | Unicode version |
Description: A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
Ref | Expression |
---|---|
nnge1d.1 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
nnap0d |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | nnap0 8944 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 1, 2 | syl 14 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-inn 8916 |
This theorem is referenced by: qtri3or 10238 qbtwnrelemcalc 10251 intfracq 10315 flqdiv 10316 modqmulnn 10337 facndiv 10712 bcn0 10728 bcn1 10731 bcm1k 10733 bcp1n 10734 bcp1nk 10735 bcval5 10736 bcpasc 10739 permnn 10744 divcnv 11498 trireciplem 11501 trirecip 11502 expcnvap0 11503 geo2sum 11515 geo2lim 11517 cvgratnnlemfm 11530 cvgratnnlemrate 11531 mertenslemi1 11536 eftabs 11657 efcllemp 11659 ege2le3 11672 efcj 11674 efaddlem 11675 eftlub 11691 eirraplem 11777 dvdsflip 11849 dvdsgcdidd 11987 mulgcd 12009 gcddiv 12012 sqgcd 12022 lcmgcdlem 12069 qredeu 12089 prmind2 12112 isprm5lem 12133 divgcdodd 12135 sqrt2irrlem 12153 oddpwdclemxy 12161 oddpwdclemodd 12164 oddpwdclemdc 12165 sqrt2irraplemnn 12171 sqrt2irrap 12172 qmuldeneqnum 12187 divnumden 12188 numdensq 12194 hashdvds 12213 phiprmpw 12214 pythagtriplem19 12274 pcprendvds2 12283 pcpremul 12285 pceulem 12286 pceu 12287 pcdiv 12294 pcqmul 12295 pcid 12315 pc2dvds 12321 dvdsprmpweqle 12328 pcaddlem 12330 pcadd 12331 oddprmdvds 12344 pockthlem 12346 4sqlem5 12372 mul4sqlem 12383 logbgcd1irraplemexp 14257 logbgcd1irraplemap 14258 2sqlem3 14324 2sqlem8 14330 cvgcmp2nlemabs 14640 redcwlpolemeq1 14662 |
Copyright terms: Public domain | W3C validator |