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Theorem eluni 3651
Description: Membership in class union. (Contributed by NM, 22-May-1994.)
Assertion
Ref Expression
eluni  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem eluni
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2630 . 2  |-  ( A  e.  U. B  ->  A  e.  _V )
2 elex 2630 . . . 4  |-  ( A  e.  x  ->  A  e.  _V )
32adantr 270 . . 3  |-  ( ( A  e.  x  /\  x  e.  B )  ->  A  e.  _V )
43exlimiv 1534 . 2  |-  ( E. x ( A  e.  x  /\  x  e.  B )  ->  A  e.  _V )
5 eleq1 2150 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
65anbi1d 453 . . . 4  |-  ( y  =  A  ->  (
( y  e.  x  /\  x  e.  B
)  <->  ( A  e.  x  /\  x  e.  B ) ) )
76exbidv 1753 . . 3  |-  ( y  =  A  ->  ( E. x ( y  e.  x  /\  x  e.  B )  <->  E. x
( A  e.  x  /\  x  e.  B
) ) )
8 df-uni 3649 . . 3  |-  U. B  =  { y  |  E. x ( y  e.  x  /\  x  e.  B ) }
97, 8elab2g 2760 . 2  |-  ( A  e.  _V  ->  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) ) )
101, 4, 9pm5.21nii 655 1  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   U.cuni 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-uni 3649
This theorem is referenced by:  eluni2  3652  elunii  3653  eluniab  3660  uniun  3667  uniin  3668  uniss  3669  unissb  3678  dftr2  3930  unidif0  3994  unipw  4035  uniex2  4254  uniuni  4264  limom  4418  dmuni  4634  fununi  5068  nfvres  5321  elunirn  5527  tfrlem7  6064  tfrexlem  6081  tfrcldm  6110  bj-uniex2  11464
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