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Theorem dfin4 3738
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 3718 . . 3  |-  ( A  i^i  B )  C_  A
2 dfss4 3732 . . 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 3735 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
54difeq2i 3619 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  \  ( A  \  B ) )
63, 5eqtr3i 2498 1  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3473    i^i cin 3475    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490
This theorem is referenced by:  indif  3740  cnvin  5413  imain  5664  resin  5837  elcls  19368  cmmbl  21708  mbfeqalem  21812  itg1addlem4  21869  itg1addlem5  21870  inelsiga  27803  mblfinlem4  29659  ismblfin  29660  cnambfre  29668  stoweidlem50  31378
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