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Theorem elz 9024
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 2124 . . 3  |-  ( x  =  N  ->  (
x  =  0  <->  N  =  0 ) )
2 eleq1 2180 . . 3  |-  ( x  =  N  ->  (
x  e.  NN  <->  N  e.  NN ) )
3 negeq 7923 . . . 4  |-  ( x  =  N  ->  -u x  =  -u N )
43eleq1d 2186 . . 3  |-  ( x  =  N  ->  ( -u x  e.  NN  <->  -u N  e.  NN ) )
51, 2, 43orbi123d 1274 . 2  |-  ( x  =  N  ->  (
( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) 
<->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
6 df-z 9023 . 2  |-  ZZ  =  { x  e.  RR  |  ( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) }
75, 6elrab2 2816 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 946    = wceq 1316    e. wcel 1465   RRcr 7587   0cc0 7588   -ucneg 7902   NNcn 8688   ZZcz 9022
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745  df-neg 7904  df-z 9023
This theorem is referenced by:  nnnegz  9025  zre  9026  elnnz  9032  0z  9033  elnn0z  9035  elznn0nn  9036  elznn0  9037  elznn  9038  znegcl  9053  zaddcl  9062  ztri3or0  9064  zeo  9124  addmodlteq  10139
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