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Theorem elz 9007
Description: Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
elz  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )

Proof of Theorem elz
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2122 . . 3  |-  ( x  =  N  ->  (
x  =  0  <->  N  =  0 ) )
2 eleq1 2178 . . 3  |-  ( x  =  N  ->  (
x  e.  NN  <->  N  e.  NN ) )
3 negeq 7919 . . . 4  |-  ( x  =  N  ->  -u x  =  -u N )
43eleq1d 2184 . . 3  |-  ( x  =  N  ->  ( -u x  e.  NN  <->  -u N  e.  NN ) )
51, 2, 43orbi123d 1272 . 2  |-  ( x  =  N  ->  (
( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) 
<->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
6 df-z 9006 . 2  |-  ZZ  =  { x  e.  RR  |  ( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) }
75, 6elrab2 2814 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ w3o 944    = wceq 1314    e. wcel 1463   RRcr 7583   0cc0 7584   -ucneg 7898   NNcn 8677   ZZcz 9005
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-rab 2400  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743  df-neg 7900  df-z 9006
This theorem is referenced by:  nnnegz  9008  zre  9009  elnnz  9015  0z  9016  elnn0z  9018  elznn0nn  9019  elznn0  9020  elznn  9021  znegcl  9036  zaddcl  9045  ztri3or0  9047  zeo  9107  addmodlteq  10111
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