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Theorem elvv 4764
Description: Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elvv  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
Distinct variable group:    x, y, A

Proof of Theorem elvv
StepHypRef Expression
1 elxp 4722 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
2 vex 2804 . . . . 5  |-  x  e. 
_V
3 vex 2804 . . . . 5  |-  y  e. 
_V
42, 3pm3.2i 441 . . . 4  |-  ( x  e.  _V  /\  y  e.  _V )
54biantru 491 . . 3  |-  ( A  =  <. x ,  y
>. 
<->  ( A  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
652exbii 1573 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>. 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
71, 6bitr4i 243 1  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703
This theorem is referenced by:  elvvv  4765  elvvuni  4766  ssrel  4792  elrel  4805  relop  4850  elreldm  4919  dmsnn0  5154  1stval2  6153  2ndval2  6154  1st2val  6161  2nd2val  6162  dfopab2  6190  dfoprab3s  6191  copsex2gb  6196  dftpos4  6269  tpostpos  6270  fundmen  6950  dfdm5  24203  dfrn5  24204  brtxp2  24492  pprodss4v  24495  brpprod3a  24497  brimg  24547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711
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