MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elz Structured version   Unicode version

Theorem elz 10862
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 2458 . . 3  |-  ( x  =  N  ->  (
x  =  0  <->  N  =  0 ) )
2 eleq1 2526 . . 3  |-  ( x  =  N  ->  (
x  e.  NN  <->  N  e.  NN ) )
3 negeq 9803 . . . 4  |-  ( x  =  N  ->  -u x  =  -u N )
43eleq1d 2523 . . 3  |-  ( x  =  N  ->  ( -u x  e.  NN  <->  -u N  e.  NN ) )
51, 2, 43orbi123d 1296 . 2  |-  ( x  =  N  ->  (
( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) 
<->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
6 df-z 10861 . 2  |-  ZZ  =  { x  e.  RR  |  ( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) }
75, 6elrab2 3256 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 367    \/ w3o 970    = wceq 1398    e. wcel 1823   RRcr 9480   0cc0 9481   -ucneg 9797   NNcn 10531   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-neg 9799  df-z 10861
This theorem is referenced by:  nnnegz  10863  zre  10864  elnnz  10870  0z  10871  elznn0nn  10874  elznn0  10875  elznn  10876  znegcl  10895  zeo  10944  ostthlem1  24010  ostth3  24021  elzdif0  28195  qqhval2lem  28196
  Copyright terms: Public domain W3C validator