ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elnnz Unicode version
Theorem elnnz 9456
Description: Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
elnnz |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Proof of Theorem elnnz
StepHypRef Expression
1 nnre 9117 . . . 4 |- ( N e. NN -> N e. RR )
2 orc 717 . . . 4 |- ( N e. NN -> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
3 nngt0 9135 . . . 4 |- ( N e. NN -> 0 < N )
41, 2, 3jca31 309 . . 3 |- ( N e. NN -> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
5 idd 21 . . . . . . 7 |- ( ( N e. RR /\ 0 < N ) -> ( N e. NN -> N e. NN ) )
6 lt0neg2 8616 . . . . . . . . . . . 12 |- ( N e. RR -> ( 0 < N <-> -u N < 0 ) )
7 renegcl 8407 . . . . . . . . . . . . 13 |- ( N e. RR -> -u N e. RR )
8 0re 8146 . . . . . . . . . . . . 13 |- 0 e. RR
9 ltnsym 8232 . . . . . . . . . . . . 13 |- ( ( -u N e. RR /\ 0 e. RR ) -> ( -u N < 0 -> -. 0 < -u N ) )
107, 8, 9sylancl 413 . . . . . . . . . . . 12 |- ( N e. RR -> ( -u N < 0 -> -. 0 < -u N ) )
116, 10sylbid 150 . . . . . . . . . . 11 |- ( N e. RR -> ( 0 < N -> -. 0 < -u N ) )
1211imp 124 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> -. 0 < -u N )
13 nngt0 9135 . . . . . . . . . 10 |- ( -u N e. NN -> 0 < -u N )
1412, 13nsyl 631 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. -u N e. NN )
15 gt0ne0 8574 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> N =/= 0 )
1615neneqd 2421 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. N = 0 )
17 ioran 757 . . . . . . . . 9 |- ( -. ( -u N e. NN \/ N = 0 ) <-> ( -. -u N e. NN /\ -. N = 0 ) )
1814, 16, 17sylanbrc 417 . . . . . . . 8 |- ( ( N e. RR /\ 0 < N ) -> -. ( -u N e. NN \/ N = 0 ) )
1918pm2.21d 622 . . . . . . 7 |- ( ( N e. RR /\ 0 < N ) -> ( ( -u N e. NN \/ N = 0 ) -> N e. NN ) )
205, 19jaod 722 . . . . . 6 |- ( ( N e. RR /\ 0 < N ) -> ( ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) -> N e. NN ) )
2120ex 115 . . . . 5 |- ( N e. RR -> ( 0 < N -> ( ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) -> N e. NN ) ) )
2221com23 78 . . . 4 |- ( N e. RR -> ( ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) -> ( 0 < N -> N e. NN ) ) )
2322imp31 256 . . 3 |- ( ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) -> N e. NN )
244, 23impbii 126 . 2 |- ( N e. NN <-> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
25 elz 9448 . . . 4 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
26 3orrot 1008 . . . . . 6 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ -u N e. NN \/ N = 0 ) )
27 3orass 1005 . . . . . 6 |- ( ( N e. NN \/ -u N e. NN \/ N = 0 ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
2826, 27bitri 184 . . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
2928anbi2i 457 . . . 4 |- ( ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) <-> ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) )
3025, 29bitri 184 . . 3 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) )
3130anbi1i 458 . 2 |- ( ( N e. ZZ /\ 0 < N ) <-> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
3224, 31bitr4i 187 1 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   \/ w3o 1001   = wceq 1395   e. wcel 2200   class class class wbr 4083   RRcr 7998   0cc0 7999   < clt 8181   -ucneg 8318   NNcn 9110   ZZcz 9446
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115
This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-z 9447
This theorem is referenced by:  nnssz  9463  elnnz1  9469  znnsub  9498  nn0ge0div  9534  msqznn  9547  elpq  9844  elfz1b  10286  lbfzo0  10381  fzo1fzo0n0  10383  elfzo0z  10384  fzofzim  10388  elfzodifsumelfzo  10407  exp3val  10763  nnesq  10881  swrdlsw  11201  pfxccatin12lem3  11264  nnabscl  11611  cvgratnnlemabsle  12038  p1modz1  12305  nndivdvds  12307  zdvdsdc  12323  oddge22np1  12392  evennn2n  12394  nno  12417  nnoddm1d2  12421  divalglemex  12433  divalglemeuneg  12434  divalg  12435  ndvdsadd  12442  bitsfzolem  12465  sqgcd  12550  qredeu  12619  prmind2  12642  sqrt2irrlem  12683  sqrt2irrap  12702  qgt0numnn  12721  oddprm  12782  pythagtriplem6  12793  pythagtriplem11  12797  pythagtriplem13  12799  pythagtriplem19  12805  pc2dvds  12853  pcadd  12863  4sqlem11  12924  4sqlem12  12925  mulgval  13659  mulgfng  13661  subgmulg  13725  znidomb  14622  sgmnncl  15662  mersenne  15671  gausslemma2dlem1a  15737  lgseisenlem1  15749  lgsquadlem1  15756  lgsquadlem2  15757  2sqlem8  15802
  Copyright terms: Public domain W3C validator