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Theorem eluni 3611
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 2583 . 2  |-  ( A  e.  U. B  ->  A  e.  _V )
2 elex 2583 . . . 4  |-  ( A  e.  x  ->  A  e.  _V )
32adantr 265 . . 3  |-  ( ( A  e.  x  /\  x  e.  B )  ->  A  e.  _V )
43exlimiv 1505 . 2  |-  ( E. x ( A  e.  x  /\  x  e.  B )  ->  A  e.  _V )
5 eleq1 2116 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
65anbi1d 446 . . . 4  |-  ( y  =  A  ->  (
( y  e.  x  /\  x  e.  B
)  <->  ( A  e.  x  /\  x  e.  B ) ) )
76exbidv 1722 . . 3  |-  ( y  =  A  ->  ( E. x ( y  e.  x  /\  x  e.  B )  <->  E. x
( A  e.  x  /\  x  e.  B
) ) )
8 df-uni 3609 . . 3  |-  U. B  =  { y  |  E. x ( y  e.  x  /\  x  e.  B ) }
97, 8elab2g 2712 . 2  |-  ( A  e.  _V  ->  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) ) )
101, 4, 9pm5.21nii 630 1  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   _Vcvv 2574   U.cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-uni 3609
This theorem is referenced by:  eluni2  3612  elunii  3613  eluniab  3620  uniun  3627  uniin  3628  uniss  3629  unissb  3638  dftr2  3884  unidif0  3948  unipw  3981  uniex2  4201  uniuni  4211  limom  4364  dmuni  4573  fununi  4995  nfvres  5234  elunirn  5433  tfrlem7  5964  tfrexlem  5979  bj-uniex2  10423
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