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Theorem difdif2ss 3407
Description: Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
Assertion
Ref Expression
difdif2ss  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  ( A  \  ( B  \  C ) )

Proof of Theorem difdif2ss
StepHypRef Expression
1 inssdif 3386 . . . 4  |-  ( A  i^i  C )  C_  ( A  \  ( _V  \  C ) )
2 unss2 3321 . . . 4  |-  ( ( A  i^i  C ) 
C_  ( A  \ 
( _V  \  C
) )  ->  (
( A  \  B
)  u.  ( A  i^i  C ) ) 
C_  ( ( A 
\  B )  u.  ( A  \  ( _V  \  C ) ) ) )
31, 2ax-mp 5 . . 3  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  (
( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
4 difindiss 3404 . . 3  |-  ( ( A  \  B )  u.  ( A  \ 
( _V  \  C
) ) )  C_  ( A  \  ( B  i^i  ( _V  \  C ) ) )
53, 4sstri 3179 . 2  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  ( A  \  ( B  i^i  ( _V  \  C ) ) )
6 invdif 3392 . . . 4  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
76eqcomi 2193 . . 3  |-  ( B 
\  C )  =  ( B  i^i  ( _V  \  C ) )
87difeq2i 3265 . 2  |-  ( A 
\  ( B  \  C ) )  =  ( A  \  ( B  i^i  ( _V  \  C ) ) )
95, 8sseqtrri 3205 1  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  ( A  \  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2752    \ cdif 3141    u. cun 3142    i^i cin 3143    C_ wss 3144
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157
This theorem is referenced by: (None)
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