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Theorem epelg 4497
Description: The epsilon relation and membership are the same. General version of epel 4499. (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 4215 . . . 4  |-  ( A  _E  B  <->  <. A ,  B >.  e.  _E  )
2 elopab 4464 . . . . . 6  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  <->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  x  e.  y ) )
3 vex 2961 . . . . . . . . . . 11  |-  x  e. 
_V
4 vex 2961 . . . . . . . . . . 11  |-  y  e. 
_V
53, 4pm3.2i 443 . . . . . . . . . 10  |-  ( x  e.  _V  /\  y  e.  _V )
6 opeqex 4449 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ( A  e. 
_V  /\  B  e.  _V )  <->  ( x  e. 
_V  /\  y  e.  _V ) ) )
75, 6mpbiri 226 . . . . . . . . 9  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( A  e.  _V  /\  B  e.  _V )
)
87simpld 447 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  A  e.  _V )
98adantr 453 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
109exlimivv 1646 . . . . . 6  |-  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
112, 10sylbi 189 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  ->  A  e. 
_V )
12 df-eprel 4496 . . . . 5  |-  _E  =  { <. x ,  y
>.  |  x  e.  y }
1311, 12eleq2s 2530 . . . 4  |-  ( <. A ,  B >.  e.  _E  ->  A  e.  _V )
141, 13sylbi 189 . . 3  |-  ( A  _E  B  ->  A  e.  _V )
1514a1i 11 . 2  |-  ( B  e.  V  ->  ( A  _E  B  ->  A  e.  _V ) )
16 elex 2966 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
1716a1i 11 . 2  |-  ( B  e.  V  ->  ( A  e.  B  ->  A  e.  _V ) )
18 eleq12 2500 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  e.  y  <-> 
A  e.  B ) )
1918, 12brabga 4471 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _E  B  <->  A  e.  B ) )
2019expcom 426 . 2  |-  ( B  e.  V  ->  ( A  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) ) )
2115, 17, 20pm5.21ndd 345 1  |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4214   {copab 4267    _E cep 4494
This theorem is referenced by:  epelc  4498  efrirr  4565  efrn2lp  4566  epne3  4763  cnfcomlem  7658  fpwwe2lem6  8512  ltpiord  8766  orvcelval  24728  predep  25469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-eprel 4496
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