ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ddifss Unicode version

Theorem ddifss 3365
Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3258), 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 3169 . 2  |-  A  C_  _V
2 ssddif 3361 . 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 2730    \ cdif 3118    C_ wss 3121
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-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  df-in 3127  df-ss 3134
This theorem is referenced by:  ssindif0im  3474  difdifdirss  3499
  Copyright terms: Public domain W3C validator