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Theorem inssddif 3285
Description: Intersection of two classes and class difference. In classical logic, such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
Assertion
Ref Expression
inssddif  |-  ( A  i^i  B )  C_  ( A  \  ( A  \  B ) )

Proof of Theorem inssddif
StepHypRef Expression
1 inss1 3264 . . 3  |-  ( A  i^i  B )  C_  A
2 ssddif 3278 . . 3  |-  ( ( A  i^i  B ) 
C_  A  <->  ( A  i^i  B )  C_  ( A  \  ( A  \ 
( A  i^i  B
) ) ) )
31, 2mpbi 144 . 2  |-  ( A  i^i  B )  C_  ( A  \  ( A  \  ( A  i^i  B ) ) )
4 difin 3281 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
54difeq2i 3159 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  \  ( A  \  B ) )
63, 5sseqtri 3099 1  |-  ( A  i^i  B )  C_  ( A  \  ( A  \  B ) )
Colors of variables: wff set class
Syntax hints:    \ cdif 3036    i^i cin 3038    C_ wss 3039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052
This theorem is referenced by: (None)
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