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Theorem ideqg 4575
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 4553 . . . . 5  |-  Rel  _I
21brrelexi 4468 . . . 4  |-  ( A  _I  B  ->  A  e.  _V )
32adantl 271 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  A  e.  _V )
4 simpl 107 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  B  e.  V )
53, 4jca 300 . 2  |-  ( ( B  e.  V  /\  A  _I  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
6 eleq1 2150 . . . . 5  |-  ( A  =  B  ->  ( A  e.  V  <->  B  e.  V ) )
76biimparc 293 . . . 4  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  V )
8 elex 2630 . . . 4  |-  ( A  e.  V  ->  A  e.  _V )
97, 8syl 14 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  _V )
10 simpl 107 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  B  e.  V )
119, 10jca 300 . 2  |-  ( ( B  e.  V  /\  A  =  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
12 eqeq1 2094 . . 3  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
13 eqeq2 2097 . . 3  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
14 df-id 4111 . . 3  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
1512, 13, 14brabg 4087 . 2  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _I  B  <->  A  =  B ) )
165, 11, 15pm5.21nd 863 1  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3837    _I cid 4106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435
This theorem is referenced by:  ideq  4576  ididg  4577  poleloe  4818
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