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Theorem elznn0nn 8316
Description: Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
Assertion
Ref Expression
elznn0nn  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )

Proof of Theorem elznn0nn
StepHypRef Expression
1 elz 8304 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 andi 742 . . 3  |-  ( ( N  e.  RR  /\  ( ( N  =  0  \/  N  e.  NN )  \/  -u N  e.  NN ) )  <->  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN )
)  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
3 df-3or 897 . . . 4  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( ( N  =  0  \/  N  e.  NN )  \/  -u N  e.  NN ) )
43anbi2i 438 . . 3  |-  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )  <->  ( N  e.  RR  /\  ( ( N  =  0  \/  N  e.  NN )  \/  -u N  e.  NN ) ) )
5 nn0re 8248 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  RR )
65pm4.71ri 378 . . . . 5  |-  ( N  e.  NN0  <->  ( N  e.  RR  /\  N  e. 
NN0 ) )
7 elnn0 8241 . . . . . . 7  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
8 orcom 657 . . . . . . 7  |-  ( ( N  e.  NN  \/  N  =  0 )  <-> 
( N  =  0  \/  N  e.  NN ) )
97, 8bitri 177 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  =  0  \/  N  e.  NN ) )
109anbi2i 438 . . . . 5  |-  ( ( N  e.  RR  /\  N  e.  NN0 )  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN ) ) )
116, 10bitri 177 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN ) ) )
1211orbi1i 690 . . 3  |-  ( ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  <->  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN )
)  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
132, 4, 123bitr4i 205 . 2  |-  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
141, 13bitri 177 1  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    \/ wo 639    \/ w3o 895    = wceq 1259    e. wcel 1409   RRcr 6946   0cc0 6947   -ucneg 7246   NNcn 7990   NN0cn0 8239   ZZcz 8302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-i2m1 7047  ax-rnegex 7051
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303
This theorem is referenced by:  peano2z  8338  zindd  8415  expcl2lemap  9432  mulexpzap  9460  expaddzap  9464  expmulzap  9466  absexpzap  9907
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