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Theorem ssddif 3369
Description: Double complement and subset. Similar to ddifss 3373 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 321 . . . . 5  |-  ( ( x  e.  A  ->  x  e.  B )  ->  ( x  e.  A  ->  ( x  e.  B  /\  x  e.  A
) ) )
2 simpr 110 . . . . . . . 8  |-  ( ( x  e.  B  /\  -.  x  e.  A
)  ->  -.  x  e.  A )
32con2i 627 . . . . . . 7  |-  ( x  e.  A  ->  -.  ( x  e.  B  /\  -.  x  e.  A
) )
43anim2i 342 . . . . . 6  |-  ( ( x  e.  B  /\  x  e.  A )  ->  ( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
5 eldif 3138 . . . . . . 7  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  x  e.  ( B  \  A
) ) )
6 eldif 3138 . . . . . . . . 9  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76notbii 668 . . . . . . . 8  |-  ( -.  x  e.  ( B 
\  A )  <->  -.  (
x  e.  B  /\  -.  x  e.  A
) )
87anbi2i 457 . . . . . . 7  |-  ( ( x  e.  B  /\  -.  x  e.  ( B  \  A ) )  <-> 
( x  e.  B  /\  -.  ( x  e.  B  /\  -.  x  e.  A ) ) )
95, 8bitri 184 . . . . . 6  |-  ( x  e.  ( B  \ 
( B  \  A
) )  <->  ( x  e.  B  /\  -.  (
x  e.  B  /\  -.  x  e.  A
) ) )
104, 9sylibr 134 . . . . 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 3257 . . . . 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 126 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  <->  ( x  e.  A  ->  x  e.  ( B  \  ( B  \  A
) ) ) )
1514albii 1470 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
16 dfss2 3144 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
17 dfss2 3144 . 2  |-  ( A 
C_  ( B  \ 
( B  \  A
) )  <->  A. x
( x  e.  A  ->  x  e.  ( B 
\  ( B  \  A ) ) ) )
1815, 16, 173bitr4i 212 1  |-  ( A 
C_  B  <->  A  C_  ( B  \  ( B  \  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351    e. wcel 2148    \ cdif 3126    C_ wss 3129
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142
This theorem is referenced by:  ddifss  3373  inssddif  3376
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