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Theorem ideqg 4796
Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ideqg  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )

Proof of Theorem ideqg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 4774 . . . . 5  |-  Rel  _I
21brrelex1i 4687 . . . 4  |-  ( A  _I  B  ->  A  e.  _V )
32adantl 277 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  A  e.  _V )
4 simpl 109 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  B  e.  V )
53, 4jca 306 . 2  |-  ( ( B  e.  V  /\  A  _I  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
6 eleq1 2252 . . . . 5  |-  ( A  =  B  ->  ( A  e.  V  <->  B  e.  V ) )
76biimparc 299 . . . 4  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  V )
8 elex 2763 . . . 4  |-  ( A  e.  V  ->  A  e.  _V )
97, 8syl 14 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  _V )
10 simpl 109 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  B  e.  V )
119, 10jca 306 . 2  |-  ( ( B  e.  V  /\  A  =  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
12 eqeq1 2196 . . 3  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
13 eqeq2 2199 . . 3  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
14 df-id 4311 . . 3  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
1512, 13, 14brabg 4287 . 2  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _I  B  <->  A  =  B ) )
165, 11, 15pm5.21nd 917 1  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752   class class class wbr 4018    _I cid 4306
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-ral 2473  df-rex 2474  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-id 4311  df-xp 4650  df-rel 4651
This theorem is referenced by:  ideq  4797  ididg  4798  poleloe  5046
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