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Theorem epelg 4781
Description: The epsilon relation and membership are the same. General version of epel 4783. (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 4440 . . . 4  |-  ( A  _E  B  <->  <. A ,  B >.  e.  _E  )
2 elopab 4744 . . . . . 6  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  <->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  x  e.  y ) )
3 vex 3109 . . . . . . . . . . 11  |-  x  e. 
_V
4 vex 3109 . . . . . . . . . . 11  |-  y  e. 
_V
53, 4pm3.2i 453 . . . . . . . . . 10  |-  ( x  e.  _V  /\  y  e.  _V )
6 opeqex 4727 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ( A  e. 
_V  /\  B  e.  _V )  <->  ( x  e. 
_V  /\  y  e.  _V ) ) )
75, 6mpbiri 233 . . . . . . . . 9  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( A  e.  _V  /\  B  e.  _V )
)
87simpld 457 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  A  e.  _V )
98adantr 463 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
109exlimivv 1728 . . . . . 6  |-  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
112, 10sylbi 195 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  ->  A  e. 
_V )
12 df-eprel 4780 . . . . 5  |-  _E  =  { <. x ,  y
>.  |  x  e.  y }
1311, 12eleq2s 2562 . . . 4  |-  ( <. A ,  B >.  e.  _E  ->  A  e.  _V )
141, 13sylbi 195 . . 3  |-  ( A  _E  B  ->  A  e.  _V )
1514a1i 11 . 2  |-  ( B  e.  V  ->  ( A  _E  B  ->  A  e.  _V ) )
16 elex 3115 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
1716a1i 11 . 2  |-  ( B  e.  V  ->  ( A  e.  B  ->  A  e.  _V ) )
18 eleq12 2530 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  e.  y  <-> 
A  e.  B ) )
1918, 12brabga 4750 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _E  B  <->  A  e.  B ) )
2019expcom 433 . 2  |-  ( B  e.  V  ->  ( A  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) ) )
2115, 17, 20pm5.21ndd 352 1  |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106   <.cop 4022   class class class wbr 4439   {copab 4496    _E cep 4778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-eprel 4780
This theorem is referenced by:  epelc  4782  efrirr  4849  efrn2lp  4850  epne3  6589  cnfcomlem  8134  cnfcomlemOLD  8142  fpwwe2lem6  9002  ltpiord  9254  orvcelval  28671  predep  29512
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