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Theorem elnnz 8730
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 8401 . . . 4  |-  ( N  e.  NN  ->  N  e.  RR )
2 orc 668 . . . 4  |-  ( N  e.  NN  ->  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0
) ) )
3 nngt0 8419 . . . 4  |-  ( N  e.  NN  ->  0  <  N )
41, 2, 3jca31 302 . . 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 7926 . . . . . . . . . . . 12  |-  ( N  e.  RR  ->  (
0  <  N  <->  -u N  <  0 ) )
7 renegcl 7722 . . . . . . . . . . . . 13  |-  ( N  e.  RR  ->  -u N  e.  RR )
8 0re 7467 . . . . . . . . . . . . 13  |-  0  e.  RR
9 ltnsym 7550 . . . . . . . . . . . . 13  |-  ( (
-u N  e.  RR  /\  0  e.  RR )  ->  ( -u N  <  0  ->  -.  0  <  -u N ) )
107, 8, 9sylancl 404 . . . . . . . . . . . 12  |-  ( N  e.  RR  ->  ( -u N  <  0  ->  -.  0  <  -u N
) )
116, 10sylbid 148 . . . . . . . . . . 11  |-  ( N  e.  RR  ->  (
0  <  N  ->  -.  0  <  -u N
) )
1211imp 122 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  0  <  -u N
)
13 nngt0 8419 . . . . . . . . . 10  |-  ( -u N  e.  NN  ->  0  <  -u N )
1412, 13nsyl 593 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  -u N  e.  NN )
15 gt0ne0 7884 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  0  <  N )  ->  N  =/=  0 )
1615neneqd 2276 . . . . . . . . 9  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  N  =  0
)
17 ioran 704 . . . . . . . . 9  |-  ( -.  ( -u N  e.  NN  \/  N  =  0 )  <->  ( -.  -u N  e.  NN  /\  -.  N  =  0
) )
1814, 16, 17sylanbrc 408 . . . . . . . 8  |-  ( ( N  e.  RR  /\  0  <  N )  ->  -.  ( -u N  e.  NN  \/  N  =  0 ) )
1918pm2.21d 584 . . . . . . 7  |-  ( ( N  e.  RR  /\  0  <  N )  -> 
( ( -u N  e.  NN  \/  N  =  0 )  ->  N  e.  NN ) )
205, 19jaod 672 . . . . . 6  |-  ( ( N  e.  RR  /\  0  <  N )  -> 
( ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) )  ->  N  e.  NN ) )
2120ex 113 . . . . 5  |-  ( N  e.  RR  ->  (
0  <  N  ->  ( ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) )  ->  N  e.  NN )
) )
2221com23 77 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) )  -> 
( 0  <  N  ->  N  e.  NN ) ) )
2322imp31 252 . . 3  |-  ( ( ( N  e.  RR  /\  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )  /\  0  <  N
)  ->  N  e.  NN )
244, 23impbii 124 . 2  |-  ( N  e.  NN  <->  ( ( N  e.  RR  /\  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0
) ) )  /\  0  <  N ) )
25 elz 8722 . . . 4  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
26 3orrot 930 . . . . . 6  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0
) )
27 3orass 927 . . . . . 6  |-  ( ( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0 )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
2826, 27bitri 182 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
2928anbi2i 445 . . . 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 182 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
3130anbi1i 446 . 2  |-  ( ( N  e.  ZZ  /\  0  <  N )  <->  ( ( N  e.  RR  /\  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0
) ) )  /\  0  <  N ) )
3224, 31bitr4i 185 1  |-  ( N  e.  NN  <->  ( N  e.  ZZ  /\  0  < 
N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    \/ w3o 923    = wceq 1289    e. wcel 1438   class class class wbr 3837   RRcr 7328   0cc0 7329    < clt 7501   -ucneg 7633   NNcn 8394   ZZcz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-z 8721
This theorem is referenced by:  nnssz  8737  elnnz1  8743  znnsub  8771  nn0ge0div  8803  msqznn  8816  elfz1b  9471  lbfzo0  9557  fzo1fzo0n0  9559  elfzo0z  9560  fzofzim  9564  elfzodifsumelfzo  9577  exp3val  9922  nnesq  10038  nnabscl  10498  nndivdvds  10884  zdvdsdc  10899  oddge22np1  10963  evennn2n  10965  nno  10988  nnoddm1d2  10992  divalglemex  11004  divalglemeuneg  11005  divalg  11006  ndvdsadd  11013  sqgcd  11100  qredeu  11161  prmind2  11184  sqrt2irrlem  11222  sqrt2irrap  11240  qgt0numnn  11259
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