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Theorem ddifnel 3103
Description: Double complement under universal class. The hypothesis is one way of expressing the idea that membership in  A is decidable. 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 3203. (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 266 . . 3  |-  ( ( x  e.  _V  /\  -.  x  e.  ( _V  \  A ) )  ->  x  e.  A
)
3 elndif 3096 . . . 4  |-  ( x  e.  A  ->  -.  x  e.  ( _V  \  A ) )
4 vex 2577 . . . 4  |-  x  e. 
_V
53, 4jctil 299 . . 3  |-  ( x  e.  A  ->  (
x  e.  _V  /\  -.  x  e.  ( _V  \  A ) ) )
62, 5impbii 121 . 2  |-  ( ( x  e.  _V  /\  -.  x  e.  ( _V  \  A ) )  <-> 
x  e.  A )
76difeqri 3092 1  |-  ( _V 
\  ( _V  \  A ) )  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574    \ cdif 2942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948
This theorem is referenced by: (None)
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