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Theorem difdif2ss 3257
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 3236 . . . 4  |-  ( A  i^i  C )  C_  ( A  \  ( _V  \  C ) )
2 unss2 3172 . . . 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 7 . . 3  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  (
( A  \  B
)  u.  ( A 
\  ( _V  \  C ) ) )
4 difindiss 3254 . . 3  |-  ( ( A  \  B )  u.  ( A  \ 
( _V  \  C
) ) )  C_  ( A  \  ( B  i^i  ( _V  \  C ) ) )
53, 4sstri 3035 . 2  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  ( A  \  ( B  i^i  ( _V  \  C ) ) )
6 invdif 3242 . . . 4  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
76eqcomi 2093 . . 3  |-  ( B 
\  C )  =  ( B  i^i  ( _V  \  C ) )
87difeq2i 3116 . 2  |-  ( A 
\  ( B  \  C ) )  =  ( A  \  ( B  i^i  ( _V  \  C ) ) )
95, 8sseqtr4i 3060 1  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  ( A  \  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2620    \ cdif 2997    u. cun 2998    i^i cin 2999    C_ wss 3000
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rab 2369  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013
This theorem is referenced by: (None)
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