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Theorem ddifstab 3130
Description: A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)
Assertion
Ref Expression
ddifstab  |-  ( ( _V  \  ( _V 
\  A ) )  =  A  <->  A. xSTAB  x  e.  A )
Distinct variable group:    x, A

Proof of Theorem ddifstab
StepHypRef Expression
1 dfcleq 2082 . 2  |-  ( ( _V  \  ( _V 
\  A ) )  =  A  <->  A. x
( x  e.  ( _V  \  ( _V 
\  A ) )  <-> 
x  e.  A ) )
2 eldif 3006 . . . . . . 7  |-  ( x  e.  ( _V  \ 
( _V  \  A
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( _V  \  A
) ) )
3 vex 2622 . . . . . . . 8  |-  x  e. 
_V
43biantrur 297 . . . . . . 7  |-  ( -.  x  e.  ( _V 
\  A )  <->  ( x  e.  _V  /\  -.  x  e.  ( _V  \  A
) ) )
5 eldif 3006 . . . . . . . . 9  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
63biantrur 297 . . . . . . . . 9  |-  ( -.  x  e.  A  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
75, 6bitr4i 185 . . . . . . . 8  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
87notbii 629 . . . . . . 7  |-  ( -.  x  e.  ( _V 
\  A )  <->  -.  -.  x  e.  A )
92, 4, 83bitr2i 206 . . . . . 6  |-  ( x  e.  ( _V  \ 
( _V  \  A
) )  <->  -.  -.  x  e.  A )
109bibi1i 226 . . . . 5  |-  ( ( x  e.  ( _V 
\  ( _V  \  A ) )  <->  x  e.  A )  <->  ( -.  -.  x  e.  A  <->  x  e.  A ) )
11 bi1 116 . . . . . 6  |-  ( ( -.  -.  x  e.  A  <->  x  e.  A
)  ->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
12 id 19 . . . . . . 7  |-  ( ( -.  -.  x  e.  A  ->  x  e.  A )  ->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
13 notnot 594 . . . . . . 7  |-  ( x  e.  A  ->  -.  -.  x  e.  A
)
1412, 13impbid1 140 . . . . . 6  |-  ( ( -.  -.  x  e.  A  ->  x  e.  A )  ->  ( -.  -.  x  e.  A  <->  x  e.  A ) )
1511, 14impbii 124 . . . . 5  |-  ( ( -.  -.  x  e.  A  <->  x  e.  A
)  <->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
1610, 15bitri 182 . . . 4  |-  ( ( x  e.  ( _V 
\  ( _V  \  A ) )  <->  x  e.  A )  <->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
17 df-stab 776 . . . 4  |-  (STAB  x  e.  A  <->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
1816, 17bitr4i 185 . . 3  |-  ( ( x  e.  ( _V 
\  ( _V  \  A ) )  <->  x  e.  A )  <-> STAB  x  e.  A
)
1918albii 1404 . 2  |-  ( A. x ( x  e.  ( _V  \  ( _V  \  A ) )  <-> 
x  e.  A )  <->  A. xSTAB  x  e.  A )
201, 19bitri 182 1  |-  ( ( _V  \  ( _V 
\  A ) )  =  A  <->  A. xSTAB  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103  STAB wstab 775   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619    \ cdif 2994
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-in1 579  ax-in2 580  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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-stab 776  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999
This theorem is referenced by: (None)
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