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Theorem epelg 4436
Description: The epsilon relation and membership are the same. General version of epel 4438. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
epelg  |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )

Proof of Theorem epelg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4154 . . . 4  |-  ( A  _E  B  <->  <. A ,  B >.  e.  _E  )
2 elopab 4403 . . . . . 6  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  <->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  x  e.  y ) )
3 vex 2902 . . . . . . . . . . 11  |-  x  e. 
_V
4 vex 2902 . . . . . . . . . . 11  |-  y  e. 
_V
53, 4pm3.2i 442 . . . . . . . . . 10  |-  ( x  e.  _V  /\  y  e.  _V )
6 opeqex 4388 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ( A  e. 
_V  /\  B  e.  _V )  <->  ( x  e. 
_V  /\  y  e.  _V ) ) )
75, 6mpbiri 225 . . . . . . . . 9  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( A  e.  _V  /\  B  e.  _V )
)
87simpld 446 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  A  e.  _V )
98adantr 452 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
109exlimivv 1642 . . . . . 6  |-  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
112, 10sylbi 188 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  ->  A  e. 
_V )
12 df-eprel 4435 . . . . 5  |-  _E  =  { <. x ,  y
>.  |  x  e.  y }
1311, 12eleq2s 2479 . . . 4  |-  ( <. A ,  B >.  e.  _E  ->  A  e.  _V )
141, 13sylbi 188 . . 3  |-  ( A  _E  B  ->  A  e.  _V )
1514a1i 11 . 2  |-  ( B  e.  V  ->  ( A  _E  B  ->  A  e.  _V ) )
16 elex 2907 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
1716a1i 11 . 2  |-  ( B  e.  V  ->  ( A  e.  B  ->  A  e.  _V ) )
18 eleq12 2449 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  e.  y  <-> 
A  e.  B ) )
1918, 12brabga 4410 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _E  B  <->  A  e.  B ) )
2019expcom 425 . 2  |-  ( B  e.  V  ->  ( A  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) ) )
2115, 17, 20pm5.21ndd 344 1  |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899   <.cop 3760   class class class wbr 4153   {copab 4206    _E cep 4433
This theorem is referenced by:  epelc  4437  efrirr  4504  efrn2lp  4505  epne3  4701  cnfcomlem  7589  fpwwe2lem6  8443  ltpiord  8697  orvcelval  24505  predep  25216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-eprel 4435
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