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Theorem ddifss 3388
Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3281), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
ddifss  |-  A  C_  ( _V  \  ( _V  \  A ) )

Proof of Theorem ddifss
StepHypRef Expression
1 ssv 3192 . 2  |-  A  C_  _V
2 ssddif 3384 . 2  |-  ( A 
C_  _V  <->  A  C_  ( _V 
\  ( _V  \  A ) ) )
31, 2mpbi 145 1  |-  A  C_  ( _V  \  ( _V  \  A ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2752    \ cdif 3141    C_ wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157
This theorem is referenced by:  ssindif0im  3497  difdifdirss  3522
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