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Theorem ideqg 4690
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 4668 . . . . 5  |-  Rel  _I
21brrelex1i 4582 . . . 4  |-  ( A  _I  B  ->  A  e.  _V )
32adantl 275 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  A  e.  _V )
4 simpl 108 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  B  e.  V )
53, 4jca 304 . 2  |-  ( ( B  e.  V  /\  A  _I  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
6 eleq1 2202 . . . . 5  |-  ( A  =  B  ->  ( A  e.  V  <->  B  e.  V ) )
76biimparc 297 . . . 4  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  V )
8 elex 2697 . . . 4  |-  ( A  e.  V  ->  A  e.  _V )
97, 8syl 14 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  _V )
10 simpl 108 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  B  e.  V )
119, 10jca 304 . 2  |-  ( ( B  e.  V  /\  A  =  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
12 eqeq1 2146 . . 3  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
13 eqeq2 2149 . . 3  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
14 df-id 4215 . . 3  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
1512, 13, 14brabg 4191 . 2  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _I  B  <->  A  =  B ) )
165, 11, 15pm5.21nd 901 1  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2686   class class class wbr 3929    _I cid 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546
This theorem is referenced by:  ideq  4691  ididg  4692  poleloe  4938
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