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Theorem ideqg 5102
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 5078 . . . . 5  |-  Rel  _I
21brrelexi 4990 . . . 4  |-  ( A  _I  B  ->  A  e.  _V )
32adantl 466 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  A  e.  _V )
4 simpl 457 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  B  e.  V )
53, 4jca 532 . 2  |-  ( ( B  e.  V  /\  A  _I  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
6 eleq1 2526 . . . . 5  |-  ( A  =  B  ->  ( A  e.  V  <->  B  e.  V ) )
76biimparc 487 . . . 4  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  V )
8 elex 3087 . . . 4  |-  ( A  e.  V  ->  A  e.  _V )
97, 8syl 16 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  _V )
10 simpl 457 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  B  e.  V )
119, 10jca 532 . 2  |-  ( ( B  e.  V  /\  A  =  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
12 eqeq1 2458 . . 3  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
13 eqeq2 2469 . . 3  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
14 df-id 4747 . . 3  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
1512, 13, 14brabg 4719 . 2  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _I  B  <->  A  =  B ) )
165, 11, 15pm5.21nd 893 1  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   class class class wbr 4403    _I cid 4742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958
This theorem is referenced by:  ideq  5103  ididg  5104  restidsing  5273  poleloe  5343  pltval  15252  tglngne  23123  tgelrnln  23178  tghilbert1_1  23185
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