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Theorem dfin4 3384
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 3364 . . 3  |-  ( A  i^i  B )  C_  A
2 dfss4 3378 . . 3  |-  ( ( A  i^i  B ) 
C_  A  <->  ( A  \  ( A  \  ( A  i^i  B ) ) )  =  ( A  i^i  B ) )
31, 2mpbi 201 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  i^i  B
)
4 difin 3381 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
54difeq2i 3266 . 2  |-  ( A 
\  ( A  \ 
( A  i^i  B
) ) )  =  ( A  \  ( A  \  B ) )
63, 5eqtr3i 2280 1  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    \ cdif 3124    i^i cin 3126    C_ wss 3127
This theorem is referenced by:  indif  3386  cnvin  5076  imain  5266  resin  5433  elcls  16773  cmmbl  18855  mbfeqalem  18960  itg1addlem4  19017  itg1addlem5  19018  stoweidlem50  27168
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rab 2527  df-v 2765  df-dif 3130  df-in 3134  df-ss 3141
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