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Theorem elopaelxp 4918
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 458 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ps )  ->  A  =  <. x ,  y >. )
212eximi 1703 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps )  ->  E. x E. y  A  =  <. x ,  y >. )
3 elopab 4720 . 2  |-  ( A  e.  { <. x ,  y >.  |  ps } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) )
4 elvv 4904 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
52, 3, 43imtr4i 269 1  |-  ( A  e.  { <. x ,  y >.  |  ps }  ->  A  e.  ( _V  X.  _V )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   _Vcvv 3078   <.cop 3999   {copab 4474    X. cxp 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-opab 4476  df-xp 4851
This theorem is referenced by:  bropaex12  4919  wlkcompim  25140  clwlkcompim  25378  linedegen  30736  opelopab3  31791
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