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Theorem elz 10746
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 2454 . . 3  |-  ( x  =  N  ->  (
x  =  0  <->  N  =  0 ) )
2 eleq1 2521 . . 3  |-  ( x  =  N  ->  (
x  e.  NN  <->  N  e.  NN ) )
3 negeq 9700 . . . 4  |-  ( x  =  N  ->  -u x  =  -u N )
43eleq1d 2519 . . 3  |-  ( x  =  N  ->  ( -u x  e.  NN  <->  -u N  e.  NN ) )
51, 2, 43orbi123d 1289 . 2  |-  ( x  =  N  ->  (
( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) 
<->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
6 df-z 10745 . 2  |-  ZZ  =  { x  e.  RR  |  ( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) }
75, 6elrab2 3213 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1758   RRcr 9379   0cc0 9380   -ucneg 9694   NNcn 10420   ZZcz 10744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-rex 2799  df-rab 2802  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-iota 5476  df-fv 5521  df-ov 6190  df-neg 9696  df-z 10745
This theorem is referenced by:  nnnegz  10747  zre  10748  elnnz  10754  0z  10755  elznn0nn  10758  elznn0  10759  elznn  10760  znegcl  10778  zeo  10825  ostthlem1  22989  ostth3  23000  elzdif0  26540  qqhval2lem  26541
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