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Theorem ideqg 5094
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 5070 . . . . 5  |-  Rel  _I
21brrelexi 4981 . . . 4  |-  ( A  _I  B  ->  A  e.  _V )
32adantl 464 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  A  e.  _V )
4 simpl 455 . . 3  |-  ( ( B  e.  V  /\  A  _I  B )  ->  B  e.  V )
53, 4jca 530 . 2  |-  ( ( B  e.  V  /\  A  _I  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
6 eleq1 2472 . . . . 5  |-  ( A  =  B  ->  ( A  e.  V  <->  B  e.  V ) )
76biimparc 485 . . . 4  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  V )
8 elex 3065 . . . 4  |-  ( A  e.  V  ->  A  e.  _V )
97, 8syl 17 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  A  e.  _V )
10 simpl 455 . . 3  |-  ( ( B  e.  V  /\  A  =  B )  ->  B  e.  V )
119, 10jca 530 . 2  |-  ( ( B  e.  V  /\  A  =  B )  ->  ( A  e.  _V  /\  B  e.  V ) )
12 eqeq1 2404 . . 3  |-  ( x  =  A  ->  (
x  =  y  <->  A  =  y ) )
13 eqeq2 2415 . . 3  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
14 df-id 4735 . . 3  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
1512, 13, 14brabg 4706 . 2  |-  ( ( A  e.  _V  /\  B  e.  V )  ->  ( A  _I  B  <->  A  =  B ) )
165, 11, 15pm5.21nd 899 1  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   _Vcvv 3056   class class class wbr 4392    _I cid 4730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947
This theorem is referenced by:  ideq  5095  ididg  5096  restidsing  5269  poleloe  5338  pltval  15804  tglngne  24215  tgelrnln  24290  opeldifid  27773  fourierdlem42  37266
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