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Theorem epelg 4308
Description: The epsilon relation and membership are the same. General version of epel 4310. (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 4019 . . . 4  |-  ( A  _E  B  <->  <. A ,  B >.  e.  _E  )
2 elopab 4276 . . . . . 6  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  <->  E. x E. y
( <. A ,  B >.  =  <. x ,  y
>.  /\  x  e.  y ) )
3 vex 2755 . . . . . . . . . . 11  |-  x  e. 
_V
4 vex 2755 . . . . . . . . . . 11  |-  y  e. 
_V
53, 4pm3.2i 272 . . . . . . . . . 10  |-  ( x  e.  _V  /\  y  e.  _V )
6 opeqex 4267 . . . . . . . . . 10  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( ( A  e. 
_V  /\  B  e.  _V )  <->  ( x  e. 
_V  /\  y  e.  _V ) ) )
75, 6mpbiri 168 . . . . . . . . 9  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  ( A  e.  _V  /\  B  e.  _V )
)
87simpld 112 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. x ,  y >.  ->  A  e.  _V )
98adantr 276 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
109exlimivv 1908 . . . . . 6  |-  ( E. x E. y (
<. A ,  B >.  = 
<. x ,  y >.  /\  x  e.  y
)  ->  A  e.  _V )
112, 10sylbi 121 . . . . 5  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  x  e.  y }  ->  A  e. 
_V )
12 df-eprel 4307 . . . . 5  |-  _E  =  { <. x ,  y
>.  |  x  e.  y }
1311, 12eleq2s 2284 . . . 4  |-  ( <. A ,  B >.  e.  _E  ->  A  e.  _V )
141, 13sylbi 121 . . 3  |-  ( A  _E  B  ->  A  e.  _V )
1514a1i 9 . 2  |-  ( B  e.  V  ->  ( A  _E  B  ->  A  e.  _V ) )
16 elex 2763 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
1716a1i 9 . 2  |-  ( B  e.  V  ->  ( A  e.  B  ->  A  e.  _V ) )
18 eleq12 2254 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  e.  y  <-> 
A  e.  B ) )
1918, 12brabga 4282 . . 3  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _E  B  <->  A  e.  B ) )
2019expcom 116 . 2  |-  ( B  e.  V  ->  ( A  e.  _V  ->  ( A  _E  B  <->  A  e.  B ) ) )
2115, 17, 20pm5.21ndd 706 1  |-  ( B  e.  V  ->  ( A  _E  B  <->  A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160   _Vcvv 2752   <.cop 3610   class class class wbr 4018   {copab 4078    _E cep 4305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-eprel 4307
This theorem is referenced by:  epelc  4309  efrirr  4371  smoiso  6328  ecidg  6626  ordiso2  7065  ltpiord  7349
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