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Theorem elnnz 9171
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 8834 . . . 4 |- ( N e. NN -> N e. RR )
2 orc 702 . . . 4 |- ( N e. NN -> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
3 nngt0 8852 . . . 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 8338 . . . . . . . . . . . 12 |- ( N e. RR -> ( 0 < N <-> -u N < 0 ) )
7 renegcl 8130 . . . . . . . . . . . . 13 |- ( N e. RR -> -u N e. RR )
8 0re 7872 . . . . . . . . . . . . 13 |- 0 e. RR
9 ltnsym 7957 . . . . . . . . . . . . 13 |- ( ( -u N e. RR /\ 0 e. RR ) -> ( -u N < 0 -> -. 0 < -u N ) )
107, 8, 9sylancl 410 . . . . . . . . . . . 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 8852 . . . . . . . . . 10 |- ( -u N e. NN -> 0 < -u N )
1412, 13nsyl 618 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. -u N e. NN )
15 gt0ne0 8296 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> N =/= 0 )
1615neneqd 2348 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. N = 0 )
17 ioran 742 . . . . . . . . 9 |- ( -. ( -u N e. NN \/ N = 0 ) <-> ( -. -u N e. NN /\ -. N = 0 ) )
1814, 16, 17sylanbrc 414 . . . . . . . 8 |- ( ( N e. RR /\ 0 < N ) -> -. ( -u N e. NN \/ N = 0 ) )
1918pm2.21d 609 . . . . . . 7 |- ( ( N e. RR /\ 0 < N ) -> ( ( -u N e. NN \/ N = 0 ) -> N e. NN ) )
205, 19jaod 707 . . . . . 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 9163 . . . 4 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
26 3orrot 969 . . . . . 6 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ -u N e. NN \/ N = 0 ) )
27 3orass 966 . . . . . 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 453 . . . 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 454 . 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 698   \/ w3o 962   = wceq 1335   e. wcel 2128   class class class wbr 3965   RRcr 7725   0cc0 7726   < clt 7906   -ucneg 8041   NNcn 8827   ZZcz 9161
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7817  ax-resscn 7818  ax-1cn 7819  ax-1re 7820  ax-icn 7821  ax-addcl 7822  ax-addrcl 7823  ax-mulcl 7824  ax-addcom 7826  ax-addass 7828  ax-distr 7830  ax-i2m1 7831  ax-0lt1 7832  ax-0id 7834  ax-rnegex 7835  ax-cnre 7837  ax-pre-ltirr 7838  ax-pre-ltwlin 7839  ax-pre-lttrn 7840  ax-pre-ltadd 7842
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-pnf 7908  df-mnf 7909  df-xr 7910  df-ltxr 7911  df-le 7912  df-sub 8042  df-neg 8043  df-inn 8828  df-z 9162
This theorem is referenced by:  nnssz  9178  elnnz1  9184  znnsub  9212  nn0ge0div  9245  msqznn  9258  elpq  9550  elfz1b  9985  lbfzo0  10073  fzo1fzo0n0  10075  elfzo0z  10076  fzofzim  10080  elfzodifsumelfzo  10093  exp3val  10414  nnesq  10530  nnabscl  10993  cvgratnnlemabsle  11417  p1modz1  11683  nndivdvds  11685  zdvdsdc  11700  oddge22np1  11764  evennn2n  11766  nno  11789  nnoddm1d2  11793  divalglemex  11805  divalglemeuneg  11806  divalg  11807  ndvdsadd  11814  sqgcd  11904  qredeu  11965  prmind2  11988  sqrt2irrlem  12026  sqrt2irrap  12045  qgt0numnn  12064
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