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Theorem ddifss 3309
Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3202), 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 3114 . 2  |-  A  C_  _V
2 ssddif 3305 . 2  |-  ( A 
C_  _V  <->  A  C_  ( _V 
\  ( _V  \  A ) ) )
31, 2mpbi 144 1  |-  A  C_  ( _V  \  ( _V  \  A ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2681    \ cdif 3063    C_ wss 3066
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079
This theorem is referenced by:  ssindif0im  3417  difdifdirss  3442
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