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Theorem elznn 9093
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 9079 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 recn 7776 . . . . . . . 8  |-  ( N  e.  RR  ->  N  e.  CC )
32negeq0d 8088 . . . . . . 7  |-  ( N  e.  RR  ->  ( N  =  0  <->  -u N  =  0 ) )
43orbi2d 780 . . . . . 6  |-  ( N  e.  RR  ->  (
( -u N  e.  NN  \/  N  =  0
)  <->  ( -u N  e.  NN  \/  -u N  =  0 ) ) )
5 elnn0 9002 . . . . . 6  |-  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  -u N  =  0 ) )
64, 5syl6rbbr 198 . . . . 5  |-  ( N  e.  RR  ->  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
76orbi2d 780 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN  \/  -u N  e.  NN0 ) 
<->  ( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
8 3orrot 969 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0
) )
9 3orass 966 . . . . 5  |-  ( ( N  e.  NN  \/  -u N  e.  NN  \/  N  =  0 )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
108, 9bitri 183 . . . 4  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  e.  NN  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
117, 10syl6rbbr 198 . . 3  |-  ( N  e.  RR  ->  (
( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) 
<->  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
1211pm5.32i 450 . 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 183 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN  \/  -u N  e.  NN0 ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 698    \/ w3o 962    = wceq 1332    e. wcel 1481   RRcr 7642   0cc0 7643   -ucneg 7957   NNcn 8743   NN0cn0 9000   ZZcz 9077
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-setind 4459  ax-resscn 7735  ax-1cn 7736  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-sub 7958  df-neg 7959  df-n0 9001  df-z 9078
This theorem is referenced by:  znnen  11945  logbgcd1irraplemexp  13091
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