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Theorem eluni2 4223
Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
eluni2  |-  ( A  e.  U. B  <->  E. x  e.  B  A  e.  x )
Distinct variable groups:    x, A    x, B

Proof of Theorem eluni2
StepHypRef Expression
1 exancom 1716 . 2  |-  ( E. x ( A  e.  x  /\  x  e.  B )  <->  E. x
( x  e.  B  /\  A  e.  x
) )
2 eluni 4222 . 2  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
3 df-rex 2777 . 2  |-  ( E. x  e.  B  A  e.  x  <->  E. x ( x  e.  B  /\  A  e.  x ) )
41, 2, 33bitr4i 280 1  |-  ( A  e.  U. B  <->  E. x  e.  B  A  e.  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370   E.wex 1657    e. wcel 1872   E.wrex 2772   U.cuni 4219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-v 3082  df-uni 4220
This theorem is referenced by:  uni0b  4244  intssuni  4278  iuncom4  4307  inuni  4586  cnvuni  5040  chfnrn  6009  ssorduni  6627  unon  6673  limuni3  6694  onfununi  7072  oarec  7275  uniinqs  7455  fissuni  7889  finsschain  7891  r1sdom  8254  rankuni2b  8333  cflm  8688  coflim  8699  axdc3lem2  8889  fpwwe2lem12  9074  uniwun  9173  tskr1om2  9201  tskuni  9216  axgroth3  9264  inaprc  9269  tskmval  9272  tskmcl  9274  suplem1pr  9485  lbsextlem2  18382  lbsextlem3  18383  isbasis3g  19963  eltg2b  19973  unitgOLD  19982  tgcl  19984  ppttop  20021  epttop  20023  neiptoptop  20146  tgcmp  20415  locfincmp  20540  dissnref  20542  comppfsc  20546  1stckgenlem  20567  txuni2  20579  txcmplem1  20655  tgqtop  20726  filuni  20899  alexsubALTlem4  21064  ptcmplem3  21068  ptcmplem4  21069  utoptop  21248  icccmplem1  21839  icccmplem3  21841  cnheibor  21982  bndth  21985  lebnumlem1  21988  lebnumlem1OLD  21991  bcthlem4  22294  ovolicc2lem5  22474  dyadmbllem  22556  itg2gt0  22717  rexunirn  28126  unipreima  28248  acunirnmpt2  28265  acunirnmpt2f  28266  reff  28675  locfinreflem  28676  cmpcref  28686  ddemeas  29068  dya2iocuni  29114  bnj1379  29651  cvmsss2  30006  cvmseu  30008  untuni  30345  dfon2lem3  30439  dfon2lem7  30443  dfon2lem8  30444  brbigcup  30671  neibastop1  31021  neibastop2lem  31022  heicant  31940  mblfinlem1  31942  cover2  32005  heiborlem9  32116  unichnidl  32229  prtlem16  32410  prter2  32422  prter3  32423  disjinfi  37430  cncfuni  37705  intsaluni  38110
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