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Theorem epelg 4306
Description: The epsilon relation and membership are the same. General version of epel 4308. (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 4024 . . . 4  |-  ( A  _E  B  <->  <. A ,  B >.  e.  _E  )
2 elopab 4272 . . . . . 6  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  <->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  x  e.  y ) )
3 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
4 vex 2791 . . . . . . . . . . 11  |-  y  e. 
_V
53, 4pm3.2i 441 . . . . . . . . . 10  |-  ( x  e.  _V  /\  y  e.  _V )
6 opeqex 4257 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ( A  e. 
_V  /\  B  e.  _V )  <->  ( x  e. 
_V  /\  y  e.  _V ) ) )
75, 6mpbiri 224 . . . . . . . . 9  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( A  e.  _V  /\  B  e.  _V )
)
87simpld 445 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  A  e.  _V )
98adantr 451 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
109exlimivv 1667 . . . . . 6  |-  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
112, 10sylbi 187 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  ->  A  e. 
_V )
12 df-eprel 4305 . . . . 5  |-  _E  =  { <. x ,  y
>.  |  x  e.  y }
1311, 12eleq2s 2375 . . . 4  |-  ( <. A ,  B >.  e.  _E  ->  A  e.  _V )
141, 13sylbi 187 . . 3  |-  ( A  _E  B  ->  A  e.  _V )
1514a1i 10 . 2  |-  ( B  e.  V  ->  ( A  _E  B  ->  A  e.  _V ) )
16 elex 2796 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
1716a1i 10 . 2  |-  ( B  e.  V  ->  ( A  e.  B  ->  A  e.  _V ) )
18 eleq12 2345 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  e.  y  <-> 
A  e.  B ) )
1918, 12brabga 4279 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _E  B  <->  A  e.  B ) )
2019expcom 424 . 2  |-  ( B  e.  V  ->  ( A  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) ) )
2115, 17, 20pm5.21ndd 343 1  |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023   {copab 4076    _E cep 4303
This theorem is referenced by:  epelc  4307  efrirr  4374  efrn2lp  4375  epne3  4572  cnfcomlem  7402  fpwwe2lem6  8257  ltpiord  8511  orvcelval  23669  predep  24192
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  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-br 4024  df-opab 4078  df-eprel 4305
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