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Theorem dfin4 3545
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 3525 . . 3  |-  ( A  i^i  B )  C_  A
2 dfss4 3539 . . 3  |-  ( ( A  i^i  B ) 
C_  A  <->  ( A  \  ( A  \  ( A  i^i  B ) ) )  =  ( A  i^i  B ) )
31, 2mpbi 200 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  i^i  B
)
4 difin 3542 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
54difeq2i 3426 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  \  ( A  \  B ) )
63, 5eqtr3i 2430 1  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    \ cdif 3281    i^i cin 3283    C_ wss 3284
This theorem is referenced by:  indif  3547  cnvin  5242  imain  5492  resin  5660  elcls  17096  cmmbl  19386  mbfeqalem  19491  itg1addlem4  19548  itg1addlem5  19549  inelsiga  24475  mblfinlem3  26149  ismblfin  26150  cnambfre  26158  stoweidlem50  27670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rab 2679  df-v 2922  df-dif 3287  df-in 3291  df-ss 3298
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