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Theorem elunii 4020
Description: Membership in class union. (Contributed by NM, 24-Mar-1995.)
Assertion
Ref Expression
elunii  |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C
)

Proof of Theorem elunii
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2497 . . . . 5  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
2 eleq1 2496 . . . . 5  |-  ( x  =  B  ->  (
x  e.  C  <->  B  e.  C ) )
31, 2anbi12d 692 . . . 4  |-  ( x  =  B  ->  (
( A  e.  x  /\  x  e.  C
)  <->  ( A  e.  B  /\  B  e.  C ) ) )
43spcegv 3037 . . 3  |-  ( B  e.  C  ->  (
( A  e.  B  /\  B  e.  C
)  ->  E. x
( A  e.  x  /\  x  e.  C
) ) )
54anabsi7 793 . 2  |-  ( ( A  e.  B  /\  B  e.  C )  ->  E. x ( A  e.  x  /\  x  e.  C ) )
6 eluni 4018 . 2  |-  ( A  e.  U. C  <->  E. x
( A  e.  x  /\  x  e.  C
) )
75, 6sylibr 204 1  |-  ( ( A  e.  B  /\  B  e.  C )  ->  A  e.  U. C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   U.cuni 4015
This theorem is referenced by:  ssuni  4037  unipw  4414  opeluu  4715  unon  4811  limuni3  4832  uniinqs  6984  trcl  7664  rankwflemb  7719  ac5num  7917  dfac3  8002  isf34lem4  8257  axcclem  8337  ttukeylem7  8395  brdom7disj  8409  brdom6disj  8410  wrdexb  11763  dprdfeq0  15580  tgss2  17052  ppttop  17071  isclo  17151  neips  17177  bwth  17473  2ndcomap  17521  2ndcsep  17522  txkgen  17684  txcon  17721  basqtop  17743  nrmr0reg  17781  alexsublem  18075  alexsubALTlem4  18081  alexsubALT  18082  ptcmplem4  18086  unirnblps  18449  unirnbl  18450  blbas  18460  met2ndci  18552  bndth  18983  dyadmbllem  19491  opnmbllem  19493  dya2iocnei  24632  dstfrvunirn  24732  pconcon  24918  cvmcov2  24962  cvmlift2lem11  25000  cvmlift2lem12  25001  onint1  26199  opnmbllem0  26242  locfincmp  26384  comppfsc  26387  neibastop2lem  26389  heibor1  26519  unichnidl  26641  prtlem16  26718  prter2  26730  stoweidlem43  27768  stoweidlem55  27780  truniALT  28626  unipwrVD  28944  unipwr  28945  truniALTVD  28990  unisnALT  29038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-uni 4016
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