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Theorem inssddif 3348
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 3327 . . 3  |-  ( A  i^i  B )  C_  A
2 ssddif 3341 . . 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 3344 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
54difeq2i 3222 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  \  ( A  \  B ) )
63, 5sseqtri 3162 1  |-  ( A  i^i  B )  C_  ( A  \  ( A  \  B ) )
Colors of variables: wff set class
Syntax hints:    \ cdif 3099    i^i cin 3101    C_ wss 3102
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115
This theorem is referenced by: (None)
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