ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eluni Unicode version

Theorem eluni 3799
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 2741 . 2  |-  ( A  e.  U. B  ->  A  e.  _V )
2 elex 2741 . . . 4  |-  ( A  e.  x  ->  A  e.  _V )
32adantr 274 . . 3  |-  ( ( A  e.  x  /\  x  e.  B )  ->  A  e.  _V )
43exlimiv 1591 . 2  |-  ( E. x ( A  e.  x  /\  x  e.  B )  ->  A  e.  _V )
5 eleq1 2233 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
65anbi1d 462 . . . 4  |-  ( y  =  A  ->  (
( y  e.  x  /\  x  e.  B
)  <->  ( A  e.  x  /\  x  e.  B ) ) )
76exbidv 1818 . . 3  |-  ( y  =  A  ->  ( E. x ( y  e.  x  /\  x  e.  B )  <->  E. x
( A  e.  x  /\  x  e.  B
) ) )
8 df-uni 3797 . . 3  |-  U. B  =  { y  |  E. x ( y  e.  x  /\  x  e.  B ) }
97, 8elab2g 2877 . 2  |-  ( A  e.  _V  ->  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) ) )
101, 4, 9pm5.21nii 699 1  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   U.cuni 3796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-uni 3797
This theorem is referenced by:  eluni2  3800  elunii  3801  eluniab  3808  uniun  3815  uniin  3816  uniss  3817  unissb  3826  dftr2  4089  unidif0  4153  unipw  4202  uniex2  4421  uniuni  4436  limom  4598  dmuni  4821  fununi  5266  nfvres  5529  elunirn  5745  tfrlem7  6296  tfrexlem  6313  tfrcldm  6342  fiuni  6955  isbasis2g  12837  tgval2  12845  ntreq0  12926  bj-uniex2  13951
  Copyright terms: Public domain W3C validator