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Theorem elvv 4936
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 4895 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
2 vex 2959 . . . . 5  |-  x  e. 
_V
3 vex 2959 . . . . 5  |-  y  e. 
_V
42, 3pm3.2i 442 . . . 4  |-  ( x  e.  _V  /\  y  e.  _V )
54biantru 492 . . 3  |-  ( A  =  <. x ,  y
>. 
<->  ( A  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
652exbii 1593 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>. 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
71, 6bitr4i 244 1  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    X. cxp 4876
This theorem is referenced by:  elvvv  4937  elvvuni  4938  ssrel  4964  elrel  4978  relop  5023  elreldm  5094  dmsnn0  5335  1stval2  6364  2ndval2  6365  1st2val  6372  2nd2val  6373  dfopab2  6401  dfoprab3s  6402  copsex2gb  6407  dftpos4  6498  tpostpos  6499  fundmen  7180  dfdm5  25400  dfrn5  25401  brtxp2  25726  pprodss4v  25729  brpprod3a  25731  brimg  25782  elopaelxp  28070
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  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-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884
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