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Theorem elopab 4243
Description: Membership in a class abstraction of ordered 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 2741 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  _V )
2 vex 2733 . . . . . 6  |-  x  e. 
_V
3 vex 2733 . . . . . 6  |-  y  e. 
_V
42, 3opex 4214 . . . . 5  |-  <. x ,  y >.  e.  _V
5 eleq1 2233 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  ( A  e. 
_V 
<-> 
<. x ,  y >.  e.  _V ) )
64, 5mpbiri 167 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  A  e.  _V )
76adantr 274 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
87exlimivv 1889 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
9 eqeq1 2177 . . . . 5  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
109anbi1d 462 . . . 4  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ph ) ) )
11102exbidv 1861 . . 3  |-  ( z  =  A  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
12 df-opab 4051 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
1311, 12elab2g 2877 . 2  |-  ( A  e.  _V  ->  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
141, 8, 13pm5.21nii 699 1  |-  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   <.cop 3586   {copab 4049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051
This theorem is referenced by:  opelopabsbALT  4244  opelopabsb  4245  opelopabt  4247  opelopabga  4248  opabm  4265  iunopab  4266  epelg  4275  elxp  4628  elco  4777  elcnv  4788  dfmpt3  5320  0neqopab  5898  brabvv  5899  opabex3d  6100  opabex3  6101
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