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Theorem elznn0nn 9075
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 9063 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 andi 807 . . 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 963 . . . 4  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( ( N  =  0  \/  N  e.  NN )  \/  -u N  e.  NN ) )
43anbi2i 452 . . 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 8993 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  RR )
65pm4.71ri 389 . . . . 5  |-  ( N  e.  NN0  <->  ( N  e.  RR  /\  N  e. 
NN0 ) )
7 elnn0 8986 . . . . . . 7  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
8 orcom 717 . . . . . . 7  |-  ( ( N  e.  NN  \/  N  =  0 )  <-> 
( N  =  0  \/  N  e.  NN ) )
97, 8bitri 183 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  =  0  \/  N  e.  NN ) )
109anbi2i 452 . . . . 5  |-  ( ( N  e.  RR  /\  N  e.  NN0 )  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN ) ) )
116, 10bitri 183 . . . 4  |-  ( N  e.  NN0  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN ) ) )
1211orbi1i 752 . . 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 211 . 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 183 1  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 697    \/ w3o 961    = wceq 1331    e. wcel 1480   RRcr 7626   0cc0 7627   -ucneg 7941   NNcn 8727   NN0cn0 8984   ZZcz 9061
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-i2m1 7732  ax-rnegex 7736
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062
This theorem is referenced by:  peano2z  9097  zindd  9176  expcl2lemap  10312  mulexpzap  10340  expaddzap  10344  expmulzap  10346  absexpzap  10859
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