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Theorem elopaba 4952
Description: Membership in an ordered pair class builder. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
copsex2ga.1  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
elopaba  |-  ( A  e.  { <. x ,  y >.  |  ps } 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
Distinct variable groups:    x, y, A    ph, x, y
Allowed substitution hints:    ps( x, y)

Proof of Theorem elopaba
StepHypRef Expression
1 elopab 4709 . 2  |-  ( A  e.  { <. x ,  y >.  |  ps } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) )
2 copsex2ga.1 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
32copsex2gb 4950 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps ) 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
41, 3bitri 257 1  |-  ( A  e.  { <. x ,  y >.  |  ps } 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   <.cop 3965   {copab 4453    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845
This theorem is referenced by:  dicelvalN  34817
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