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Theorem ddifnel 3131
Description: Double complement under universal class. The hypothesis corresponds to stability of membership in 
A, which is weaker than decidability (see dcimpstab 790). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3132) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that  A is a subset of  _V  \  ( _V  \  A ), see ddifss 3237. (Contributed by Jim Kingdon, 21-Jul-2018.)
Hypothesis
Ref Expression
ddifnel.1  |-  ( -.  x  e.  ( _V 
\  A )  ->  x  e.  A )
Assertion
Ref Expression
ddifnel  |-  ( _V 
\  ( _V  \  A ) )  =  A
Distinct variable group:    x, A

Proof of Theorem ddifnel
StepHypRef Expression
1 ddifnel.1 . . . 4  |-  ( -.  x  e.  ( _V 
\  A )  ->  x  e.  A )
21adantl 271 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  ( _V  \  A ) )  ->  x  e.  A
)
3 elndif 3124 . . . 4  |-  ( x  e.  A  ->  -.  x  e.  ( _V  \  A ) )
4 vex 2622 . . . 4  |-  x  e. 
_V
53, 4jctil 305 . . 3  |-  ( x  e.  A  ->  (
x  e.  _V  /\  -.  x  e.  ( _V  \  A ) ) )
62, 5impbii 124 . 2  |-  ( ( x  e.  _V  /\  -.  x  e.  ( _V  \  A ) )  <-> 
x  e.  A )
76difeqri 3120 1  |-  ( _V 
\  ( _V  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    \ cdif 2996
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-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 3001
This theorem is referenced by: (None)
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