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Theorem elnnz 9022
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 8691 . . . 4  |-  ( N  e.  NN  ->  N  e.  RR )
2 orc 686 . . . 4  |-  ( N  e.  NN  ->  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0
) ) )
3 nngt0 8709 . . . 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 8199 . . . . . . . . . . . 12  |-  ( N  e.  RR  ->  (
0  <  N  <->  -u N  <  0 ) )
7 renegcl 7991 . . . . . . . . . . . . 13  |-  ( N  e.  RR  ->  -u N  e.  RR )
8 0re 7734 . . . . . . . . . . . . 13  |-  0  e.  RR
9 ltnsym 7818 . . . . . . . . . . . . 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 8709 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  0  <  -u N )
1412, 13nsyl 602 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  -u N  e.  NN )
15 gt0ne0 8157 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  0  <  N )  ->  N  =/=  0 )
1615neneqd 2306 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  N  =  0
)
17 ioran 726 . . . . . . . . 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 593 . . . . . . 7  |-  ( ( N  e.  RR  /\  0  <  N )  -> 
( ( -u N  e.  NN  \/  N  =  0 )  ->  N  e.  NN ) )
205, 19jaod 691 . . . . . 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 9014 . . . 4  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
26 3orrot 953 . . . . . 6  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0
) )
27 3orass 950 . . . . . 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 682    \/ w3o 946    = wceq 1316    e. wcel 1465   class class class wbr 3899   RRcr 7587   0cc0 7588    < clt 7768   -ucneg 7902   NNcn 8684   ZZcz 9012
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-z 9013
This theorem is referenced by:  nnssz  9029  elnnz1  9035  znnsub  9063  nn0ge0div  9096  msqznn  9109  elfz1b  9825  lbfzo0  9913  fzo1fzo0n0  9915  elfzo0z  9916  fzofzim  9920  elfzodifsumelfzo  9933  exp3val  10250  nnesq  10366  nnabscl  10827  cvgratnnlemabsle  11251  nndivdvds  11411  zdvdsdc  11426  oddge22np1  11490  evennn2n  11492  nno  11515  nnoddm1d2  11519  divalglemex  11531  divalglemeuneg  11532  divalg  11533  ndvdsadd  11540  sqgcd  11629  qredeu  11690  prmind2  11713  sqrt2irrlem  11751  sqrt2irrap  11769  qgt0numnn  11788
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