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Theorem ddifstab 3259
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 2164 . 2  |-  ( ( _V  \  ( _V 
\  A ) )  =  A  <->  A. x
( x  e.  ( _V  \  ( _V 
\  A ) )  <-> 
x  e.  A ) )
2 eldif 3130 . . . . . . 7  |-  ( x  e.  ( _V  \ 
( _V  \  A
) )  <->  ( x  e.  _V  /\  -.  x  e.  ( _V  \  A
) ) )
3 vex 2733 . . . . . . . 8  |-  x  e. 
_V
43biantrur 301 . . . . . . 7  |-  ( -.  x  e.  ( _V 
\  A )  <->  ( x  e.  _V  /\  -.  x  e.  ( _V  \  A
) ) )
5 eldif 3130 . . . . . . . . 9  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
63biantrur 301 . . . . . . . . 9  |-  ( -.  x  e.  A  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
75, 6bitr4i 186 . . . . . . . 8  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
87notbii 663 . . . . . . 7  |-  ( -.  x  e.  ( _V 
\  A )  <->  -.  -.  x  e.  A )
92, 4, 83bitr2i 207 . . . . . 6  |-  ( x  e.  ( _V  \ 
( _V  \  A
) )  <->  -.  -.  x  e.  A )
109bibi1i 227 . . . . 5  |-  ( ( x  e.  ( _V 
\  ( _V  \  A ) )  <->  x  e.  A )  <->  ( -.  -.  x  e.  A  <->  x  e.  A ) )
11 biimp 117 . . . . . 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 624 . . . . . . 7  |-  ( x  e.  A  ->  -.  -.  x  e.  A
)
1412, 13impbid1 141 . . . . . 6  |-  ( ( -.  -.  x  e.  A  ->  x  e.  A )  ->  ( -.  -.  x  e.  A  <->  x  e.  A ) )
1511, 14impbii 125 . . . . 5  |-  ( ( -.  -.  x  e.  A  <->  x  e.  A
)  <->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
1610, 15bitri 183 . . . 4  |-  ( ( x  e.  ( _V 
\  ( _V  \  A ) )  <->  x  e.  A )  <->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
17 df-stab 826 . . . 4  |-  (STAB  x  e.  A  <->  ( -.  -.  x  e.  A  ->  x  e.  A ) )
1816, 17bitr4i 186 . . 3  |-  ( ( x  e.  ( _V 
\  ( _V  \  A ) )  <->  x  e.  A )  <-> STAB  x  e.  A
)
1918albii 1463 . 2  |-  ( A. x ( x  e.  ( _V  \  ( _V  \  A ) )  <-> 
x  e.  A )  <->  A. xSTAB  x  e.  A )
201, 19bitri 183 1  |-  ( ( _V  \  ( _V 
\  A ) )  =  A  <->  A. xSTAB  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  STAB wstab 825   A.wal 1346    = wceq 1348    e. wcel 2141   _Vcvv 2730    \ cdif 3118
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-stab 826  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123
This theorem is referenced by: (None)
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