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Theorem elopaelxp 5072
Description: Membership in an ordered pair class builder implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Assertion
Ref Expression
elopaelxp  |-  ( A  e.  { <. x ,  y >.  |  ps }  ->  A  e.  ( _V  X.  _V )
)
Distinct variable group:    x, A, y
Allowed substitution hints:    ps( x, y)

Proof of Theorem elopaelxp
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ps )  ->  A  =  <. x ,  y >. )
212eximi 1636 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps )  ->  E. x E. y  A  =  <. x ,  y >. )
3 elopab 4755 . 2  |-  ( A  e.  { <. x ,  y >.  |  ps } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) )
4 elvv 5058 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
52, 3, 43imtr4i 266 1  |-  ( A  e.  { <. x ,  y >.  |  ps }  ->  A  e.  ( _V  X.  _V )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113   <.cop 4033   {copab 4504    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005
This theorem is referenced by:  bropaex12  5073  wlkcompim  24299  clwlkcompim  24537
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