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

Theorem elz 10285
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 2443 . . 3  |-  ( x  =  N  ->  (
x  =  0  <->  N  =  0 ) )
2 eleq1 2497 . . 3  |-  ( x  =  N  ->  (
x  e.  NN  <->  N  e.  NN ) )
3 negeq 9299 . . . 4  |-  ( x  =  N  ->  -u x  =  -u N )
43eleq1d 2503 . . 3  |-  ( x  =  N  ->  ( -u x  e.  NN  <->  -u N  e.  NN ) )
51, 2, 43orbi123d 1254 . 2  |-  ( x  =  N  ->  (
( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) 
<->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
6 df-z 10284 . 2  |-  ZZ  =  { x  e.  RR  |  ( x  =  0  \/  x  e.  NN  \/  -u x  e.  NN ) }
75, 6elrab2 3095 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726   RRcr 8990   0cc0 8991   -ucneg 9293   NNcn 10001   ZZcz 10283
This theorem is referenced by:  nnnegz  10286  zre  10287  elnnz  10293  0z  10294  elznn0nn  10296  elznn0  10297  elznn  10298  znegcl  10314  zeo  10356  ostthlem1  21322  ostth3  21333  elzdif0  24365  qqhval2lem  24366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-neg 9295  df-z 10284
  Copyright terms: Public domain W3C validator