MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elvv Unicode version

Theorem elvv 4748
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 4706 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y
( A  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
2 vex 2791 . . . . 5  |-  x  e. 
_V
3 vex 2791 . . . . 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 1570 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687
This theorem is referenced by:  elvvv  4749  elvvuni  4750  ssrel  4776  elrel  4789  relop  4834  elreldm  4903  dmsnn0  5138  1stval2  6137  2ndval2  6138  1st2val  6145  2nd2val  6146  dfopab2  6174  dfoprab3s  6175  copsex2gb  6180  dftpos4  6253  tpostpos  6254  fundmen  6934  dfdm5  23543  dfrn5  23544  brtxp2  23832  pprodss4v  23835  brpprod3a  23837  brimg  23887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
  Copyright terms: Public domain W3C validator