ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elnnz Unicode version
Theorem elnnz 9071
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 8734 . . . 4 |- ( N e. NN -> N e. RR )
2 orc 701 . . . 4 |- ( N e. NN -> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
3 nngt0 8752 . . . 4 |- ( N e. NN -> 0 < N )
41, 2, 3jca31 307 . . 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 8238 . . . . . . . . . . . 12 |- ( N e. RR -> ( 0 < N <-> -u N < 0 ) )
7 renegcl 8030 . . . . . . . . . . . . 13 |- ( N e. RR -> -u N e. RR )
8 0re 7773 . . . . . . . . . . . . 13 |- 0 e. RR
9 ltnsym 7857 . . . . . . . . . . . . 13 |- ( ( -u N e. RR /\ 0 e. RR ) -> ( -u N < 0 -> -. 0 < -u N ) )
107, 8, 9sylancl 409 . . . . . . . . . . . 12 |- ( N e. RR -> ( -u N < 0 -> -. 0 < -u N ) )
116, 10sylbid 149 . . . . . . . . . . 11 |- ( N e. RR -> ( 0 < N -> -. 0 < -u N ) )
1211imp 123 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> -. 0 < -u N )
13 nngt0 8752 . . . . . . . . . 10 |- ( -u N e. NN -> 0 < -u N )
1412, 13nsyl 617 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. -u N e. NN )
15 gt0ne0 8196 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> N =/= 0 )
1615neneqd 2329 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. N = 0 )
17 ioran 741 . . . . . . . . 9 |- ( -. ( -u N e. NN \/ N = 0 ) <-> ( -. -u N e. NN /\ -. N = 0 ) )
1814, 16, 17sylanbrc 413 . . . . . . . 8 |- ( ( N e. RR /\ 0 < N ) -> -. ( -u N e. NN \/ N = 0 ) )
1918pm2.21d 608 . . . . . . 7 |- ( ( N e. RR /\ 0 < N ) -> ( ( -u N e. NN \/ N = 0 ) -> N e. NN ) )
205, 19jaod 706 . . . . . 6 |- ( ( N e. RR /\ 0 < N ) -> ( ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) -> N e. NN ) )
2120ex 114 . . . . 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 254 . . 3 |- ( ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) -> N e. NN )
244, 23impbii 125 . 2 |- ( N e. NN <-> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
25 elz 9063 . . . 4 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
26 3orrot 968 . . . . . 6 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ -u N e. NN \/ N = 0 ) )
27 3orass 965 . . . . . 6 |- ( ( N e. NN \/ -u N e. NN \/ N = 0 ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
2826, 27bitri 183 . . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
2928anbi2i 452 . . . 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 183 . . 3 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) )
3130anbi1i 453 . 2 |- ( ( N e. ZZ /\ 0 < N ) <-> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
3224, 31bitr4i 186 1 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   \/ w3o 961   = wceq 1331   e. wcel 1480   class class class wbr 3929   RRcr 7626   0cc0 7627   < clt 7807   -ucneg 7941   NNcn 8727   ZZcz 9061
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-z 9062
This theorem is referenced by:  nnssz  9078  elnnz1  9084  znnsub  9112  nn0ge0div  9145  msqznn  9158  elfz1b  9877  lbfzo0  9965  fzo1fzo0n0  9967  elfzo0z  9968  fzofzim  9972  elfzodifsumelfzo  9985  exp3val  10302  nnesq  10418  nnabscl  10879  cvgratnnlemabsle  11303  nndivdvds  11506  zdvdsdc  11521  oddge22np1  11585  evennn2n  11587  nno  11610  nnoddm1d2  11614  divalglemex  11626  divalglemeuneg  11627  divalg  11628  ndvdsadd  11635  sqgcd  11724  qredeu  11785  prmind2  11808  sqrt2irrlem  11846  sqrt2irrap  11865  qgt0numnn  11884
  Copyright terms: Public domain W3C validator