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Theorem elopab 4015
Description: Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elopab  |-  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem elopab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 2611 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  _V )
2 vex 2605 . . . . . 6  |-  x  e. 
_V
3 vex 2605 . . . . . 6  |-  y  e. 
_V
42, 3opex 3986 . . . . 5  |-  <. x ,  y >.  e.  _V
5 eleq1 2142 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  ( A  e. 
_V 
<-> 
<. x ,  y >.  e.  _V ) )
64, 5mpbiri 166 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  A  e.  _V )
76adantr 270 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
87exlimivv 1818 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
9 eqeq1 2088 . . . . 5  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
109anbi1d 453 . . . 4  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ph ) ) )
11102exbidv 1790 . . 3  |-  ( z  =  A  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
12 df-opab 3842 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
1311, 12elab2g 2741 . 2  |-  ( A  e.  _V  ->  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
141, 8, 13pm5.21nii 653 1  |-  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602   <.cop 3403   {copab 3840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-opab 3842
This theorem is referenced by:  opelopabsbALT  4016  opelopabsb  4017  opelopabt  4019  opelopabga  4020  opabm  4037  iunopab  4038  epelg  4047  elxp  4382  elcnv  4534  dfmpt3  5046  0neqopab  5575  brabvv  5576  opabex3d  5773  opabex3  5774
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