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Theorem dfin4 3745
Description: Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
dfin4  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )

Proof of Theorem dfin4
StepHypRef Expression
1 inss1 3714 . . 3  |-  ( A  i^i  B )  C_  A
2 dfss4 3739 . . 3  |-  ( ( A  i^i  B ) 
C_  A  <->  ( A  \  ( A  \  ( A  i^i  B ) ) )  =  ( A  i^i  B ) )
31, 2mpbi 208 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  i^i  B
)
4 difin 3742 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
54difeq2i 3615 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  \  ( A  \  B ) )
63, 5eqtr3i 2488 1  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    \ cdif 3468    i^i cin 3470    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485
This theorem is referenced by:  indif  3747  cnvin  5420  imain  5670  resin  5843  elcls  19700  cmmbl  22070  mbfeqalem  22174  itg1addlem4  22231  itg1addlem5  22232  inelsiga  28296  mblfinlem4  30216  ismblfin  30217  cnambfre  30225  stoweidlem50  31993
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