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Theorem ssddif 3231
Description: Double complement and subset. Similar to ddifss 3235 but inside a class  B instead of the universal class  _V. In classical logic the subset operation on the right hand side could be an equality (that is,  A  C_  B  <->  ( B  \  ( B 
\  A ) )  =  A). (Contributed by Jim Kingdon, 24-Jul-2018.)
Assertion
Ref Expression
ssddif  |-  ( A 
C_  B  <->  A  C_  ( B  \  ( B  \  A ) ) )

Proof of Theorem ssddif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ancr 314 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( x  e.  A  ->  ( x  e.  B  /\  x  e.  A
) ) )
2 simpr 108 . . . . . . . 8  |-  ( ( x  e.  B  /\  -.  x  e.  A
)  ->  -.  x  e.  A )
32con2i 592 . . . . . . 7  |-  ( x  e.  A  ->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
43anim2i 334 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  A )  ->  ( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
5 eldif 3006 . . . . . . 7  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  x  e.  ( B  \  A
) ) )
6 eldif 3006 . . . . . . . . 9  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76notbii 629 . . . . . . . 8  |-  ( -.  x  e.  ( B 
\  A )  <->  -.  (
x  e.  B  /\  -.  x  e.  A
) )
87anbi2i 445 . . . . . . 7  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
95, 8bitri 182 . . . . . 6  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
104, 9sylibr 132 . . . . 5  |-  ( ( x  e.  B  /\  x  e.  A )  ->  x  e.  ( B 
\  ( B  \  A ) ) )
111, 10syl6 33 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
12 eldifi 3120 . . . . 5  |-  ( x  e.  ( B  \ 
( B  \  A
) )  ->  x  e.  B )
1312imim2i 12 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( B  \  ( B  \  A
) ) )  -> 
( x  e.  A  ->  x  e.  B ) )
1411, 13impbii 124 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  A  ->  x  e.  ( B  \  ( B  \  A
) ) ) )
1514albii 1404 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
16 dfss2 3012 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
17 dfss2 3012 . 2  |-  ( A 
C_  ( B  \ 
( B  \  A
) )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
1815, 16, 173bitr4i 210 1  |-  ( A 
C_  B  <->  A  C_  ( B  \  ( B  \  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    e. wcel 1438    \ cdif 2994    C_ wss 2997
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 2999  df-in 3003  df-ss 3010
This theorem is referenced by:  ddifss  3235  inssddif  3238
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