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Theorem elopab 4454
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 2956 . 2  |-  ( A  e.  { <. x ,  y >.  |  ph }  ->  A  e.  _V )
2 opex 4419 . . . . 5  |-  <. x ,  y >.  e.  _V
3 eleq1 2495 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  ( A  e. 
_V 
<-> 
<. x ,  y >.  e.  _V ) )
42, 3mpbiri 225 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  A  e.  _V )
54adantr 452 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
65exlimivv 1645 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph )  ->  A  e.  _V )
7 eqeq1 2441 . . . . 5  |-  ( z  =  A  ->  (
z  =  <. x ,  y >.  <->  A  =  <. x ,  y >.
) )
87anbi1d 686 . . . 4  |-  ( z  =  A  ->  (
( z  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ph ) ) )
982exbidv 1638 . . 3  |-  ( z  =  A  ->  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
10 df-opab 4259 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
119, 10elab2g 3076 . 2  |-  ( A  e.  _V  ->  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) ) )
121, 6, 11pm5.21nii 343 1  |-  ( A  e.  { <. x ,  y >.  |  ph } 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   {copab 4257
This theorem is referenced by:  opelopabsbOLD  4455  opelopabsb  4457  opelopabt  4459  opelopabga  4460  opabn0  4477  iunopab  4478  epelg  4487  elxp  4887  elcnv  5041  dfmpt3  5559  opabex3d  5981  opabex3  5982  0neqopab  6111  elopaba  6401  fsplit  6443  isfunc  14053  qqhval2  24358  rtrclreclem.trans  25138  pellexlem5  26877  pellex  26879  opelopab4  28565  dicelval3  31905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259
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