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Theorem elfix2 24515
Description: Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
elfix2.1  |-  Rel  R
Assertion
Ref Expression
elfix2  |-  ( A  e.  Fix R  <->  A R A )

Proof of Theorem elfix2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( A  e.  Fix R  ->  A  e.  _V )
2 elfix2.1 . . 3  |-  Rel  R
32brrelexi 4745 . 2  |-  ( A R A  ->  A  e.  _V )
4 eleq1 2356 . . 3  |-  ( x  =  A  ->  (
x  e.  Fix R  <->  A  e.  Fix R ) )
5 breq12 4044 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
65anidms 626 . . 3  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
7 vex 2804 . . . 4  |-  x  e. 
_V
87elfix 24514 . . 3  |-  ( x  e.  Fix R  <->  x R x )
94, 6, 8vtoclbg 2857 . 2  |-  ( A  e.  _V  ->  ( A  e.  Fix R  <->  A R A ) )
101, 3, 9pm5.21nii 342 1  |-  ( A  e.  Fix R  <->  A R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   Rel wrel 4710   Fixcfix 24449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-dm 4715  df-fix 24471
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