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Theorem elopab 4272
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 2796 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  _V )
2 opex 4237 . . . . 5  |-  <. x ,  y >.  e.  _V
3 eleq1 2343 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  ( A  e. 
_V 
<-> 
<. x ,  y >.  e.  _V ) )
42, 3mpbiri 224 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  A  e.  _V )
54adantr 451 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
65exlimivv 1667 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
7 eqeq1 2289 . . . . 5  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
87anbi1d 685 . . . 4  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ph ) ) )
982exbidv 1614 . . 3  |-  ( z  =  A  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
10 df-opab 4078 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
119, 10elab2g 2916 . 2  |-  ( A  e.  _V  ->  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
121, 6, 11pm5.21nii 342 1  |-  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   {copab 4076
This theorem is referenced by:  opelopabsbOLD  4273  opelopabsb  4275  opelopabt  4277  opelopabga  4278  opabn0  4295  iunopab  4296  epelg  4306  elxp  4706  elcnv  4858  dfmpt3  5366  opabex3  5769  elopaba  6182  fsplit  6223  isfunc  13738  rtrclreclem.trans  24043  cmpmor  25975  pellexlem5  26918  pellex  26920  opelopab4  28317  dicelval3  31370
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
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