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Theorem ssddif 3274
Description: Double complement and subset. Similar to ddifss 3278 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 317 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( x  e.  A  ->  ( x  e.  B  /\  x  e.  A
) ) )
2 simpr 109 . . . . . . . 8  |-  ( ( x  e.  B  /\  -.  x  e.  A
)  ->  -.  x  e.  A )
32con2i 599 . . . . . . 7  |-  ( x  e.  A  ->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
43anim2i 337 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  A )  ->  ( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
5 eldif 3044 . . . . . . 7  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  x  e.  ( B  \  A
) ) )
6 eldif 3044 . . . . . . . . 9  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76notbii 640 . . . . . . . 8  |-  ( -.  x  e.  ( B 
\  A )  <->  -.  (
x  e.  B  /\  -.  x  e.  A
) )
87anbi2i 450 . . . . . . 7  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
95, 8bitri 183 . . . . . 6  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
104, 9sylibr 133 . . . . 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 3162 . . . . 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 125 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  A  ->  x  e.  ( B  \  ( B  \  A
) ) ) )
1514albii 1427 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
16 dfss2 3050 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
17 dfss2 3050 . 2  |-  ( A 
C_  ( B  \ 
( B  \  A
) )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
1815, 16, 173bitr4i 211 1  |-  ( A 
C_  B  <->  A  C_  ( B  \  ( B  \  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1310    e. wcel 1461    \ cdif 3032    C_ wss 3035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037  df-in 3041  df-ss 3048
This theorem is referenced by:  ddifss  3278  inssddif  3281
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