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Theorem elopab 4288
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 2809 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  _V )
2 opex 4253 . . . . 5  |-  <. x ,  y >.  e.  _V
3 eleq1 2356 . . . . 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 1625 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
7 eqeq1 2302 . . . . 5  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
87anbi1d 685 . . . 4  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ph ) ) )
982exbidv 1618 . . 3  |-  ( z  =  A  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
10 df-opab 4094 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
119, 10elab2g 2929 . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   {copab 4092
This theorem is referenced by:  opelopabsbOLD  4289  opelopabsb  4291  opelopabt  4293  opelopabga  4294  opabn0  4311  iunopab  4312  epelg  4322  elxp  4722  elcnv  4874  dfmpt3  5382  opabex3  5785  elopaba  6198  fsplit  6239  isfunc  13754  rtrclreclem.trans  24058  cmpmor  26078  pellexlem5  27021  pellex  27023  opabex3d  28190  0neqopab  28192  opelopab4  28616  dicelval3  31992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094
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