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Theorem elznn 8766
Description: Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
Assertion
Ref Expression
elznn  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )

Proof of Theorem elznn
StepHypRef Expression
1 elz 8752 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 recn 7475 . . . . . . . 8  |-  ( N  e.  RR  ->  N  e.  CC )
32negeq0d 7785 . . . . . . 7  |-  ( N  e.  RR  ->  ( N  =  0  <->  -u N  =  0 ) )
43orbi2d 739 . . . . . 6  |-  ( N  e.  RR  ->  (
( -u N  e.  NN  \/  N  =  0
)  <->  ( -u N  e.  NN  \/  -u N  =  0 ) ) )
5 elnn0 8675 . . . . . 6  |-  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  -u N  =  0 ) )
64, 5syl6rbbr 197 . . . . 5  |-  ( N  e.  RR  ->  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
76orbi2d 739 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN  \/  -u N  e.  NN0 ) 
<->  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
8 3orrot 930 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0
) )
9 3orass 927 . . . . 5  |-  ( ( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0 )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
108, 9bitri 182 . . . 4  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
117, 10syl6rbbr 197 . . 3  |-  ( N  e.  RR  ->  (
( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) 
<->  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
1211pm5.32i 442 . 2  |-  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
131, 12bitri 182 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    \/ wo 664    \/ w3o 923    = wceq 1289    e. wcel 1438   RRcr 7349   0cc0 7350   -ucneg 7654   NNcn 8422   NN0cn0 8673   ZZcz 8750
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456
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-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655  df-neg 7656  df-n0 8674  df-z 8751
This theorem is referenced by:  znnen  11489
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