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Theorem difdif2ss 3228
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 3207 . . . 4  |-  ( A  i^i  C )  C_  ( A  \  ( _V  \  C ) )
2 unss2 3144 . . . 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 3225 . . 3  |-  ( ( A  \  B )  u.  ( A  \ 
( _V  \  C
) ) )  C_  ( A  \  ( B  i^i  ( _V  \  C ) ) )
53, 4sstri 3009 . 2  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  ( A  \  ( B  i^i  ( _V  \  C ) ) )
6 invdif 3213 . . . 4  |-  ( B  i^i  ( _V  \  C ) )  =  ( B  \  C
)
76eqcomi 2086 . . 3  |-  ( B 
\  C )  =  ( B  i^i  ( _V  \  C ) )
87difeq2i 3088 . 2  |-  ( A 
\  ( B  \  C ) )  =  ( A  \  ( B  i^i  ( _V  \  C ) ) )
95, 8sseqtr4i 3033 1  |-  ( ( A  \  B )  u.  ( A  i^i  C ) )  C_  ( A  \  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2602    \ cdif 2971    u. cun 2972    i^i cin 2973    C_ wss 2974
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
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