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Theorem elnnz 9015
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 8684 . . . 4  |-  ( N  e.  NN  ->  N  e.  RR )
2 orc 684 . . . 4  |-  ( N  e.  NN  ->  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0
) ) )
3 nngt0 8702 . . . 4  |-  ( N  e.  NN  ->  0  <  N )
41, 2, 3jca31 305 . . 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 8195 . . . . . . . . . . . 12  |-  ( N  e.  RR  ->  (
0  <  N  <->  -u N  <  0 ) )
7 renegcl 7987 . . . . . . . . . . . . 13  |-  ( N  e.  RR  ->  -u N  e.  RR )
8 0re 7730 . . . . . . . . . . . . 13  |-  0  e.  RR
9 ltnsym 7814 . . . . . . . . . . . . 13  |-  ( (
-u N  e.  RR  /\  0  e.  RR )  ->  ( -u N  <  0  ->  -.  0  <  -u N ) )
107, 8, 9sylancl 407 . . . . . . . . . . . 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 8702 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  0  <  -u N )
1412, 13nsyl 600 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  -u N  e.  NN )
15 gt0ne0 8153 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  0  <  N )  ->  N  =/=  0 )
1615neneqd 2304 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  N  =  0
)
17 ioran 724 . . . . . . . . 9  |-  ( -.  ( -u N  e.  NN  \/  N  =  0 )  <->  ( -.  -u N  e.  NN  /\  -.  N  =  0
) )
1814, 16, 17sylanbrc 411 . . . . . . . 8  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  ( -u N  e.  NN  \/  N  =  0 ) )
1918pm2.21d 591 . . . . . . 7  |-  ( ( N  e.  RR  /\  0  <  N )  -> 
( ( -u N  e.  NN  \/  N  =  0 )  ->  N  e.  NN ) )
205, 19jaod 689 . . . . . 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 9007 . . . 4  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
26 3orrot 951 . . . . . 6  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0
) )
27 3orass 948 . . . . . 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 450 . . . 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 451 . 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 680    \/ w3o 944    = wceq 1314    e. wcel 1463   class class class wbr 3897   RRcr 7583   0cc0 7584    < clt 7764   -ucneg 7898   NNcn 8677   ZZcz 9005
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8678  df-z 9006
This theorem is referenced by:  nnssz  9022  elnnz1  9028  znnsub  9056  nn0ge0div  9089  msqznn  9102  elfz1b  9810  lbfzo0  9898  fzo1fzo0n0  9900  elfzo0z  9901  fzofzim  9905  elfzodifsumelfzo  9918  exp3val  10235  nnesq  10351  nnabscl  10812  cvgratnnlemabsle  11236  nndivdvds  11395  zdvdsdc  11410  oddge22np1  11474  evennn2n  11476  nno  11499  nnoddm1d2  11503  divalglemex  11515  divalglemeuneg  11516  divalg  11517  ndvdsadd  11524  sqgcd  11613  qredeu  11674  prmind2  11697  sqrt2irrlem  11735  sqrt2irrap  11753  qgt0numnn  11772
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