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Theorem symdif1 3763
Description: Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
symdif1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  ( ( A  u.  B )  \  ( A  i^i  B ) )

Proof of Theorem symdif1
StepHypRef Expression
1 difundir 3751 . 2  |-  ( ( A  u.  B ) 
\  ( A  i^i  B ) )  =  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )
2 difin 3735 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
3 incom 3691 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43difeq2i 3619 . . . 4  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  ( B  i^i  A ) )
5 difin 3735 . . . 4  |-  ( B 
\  ( B  i^i  A ) )  =  ( B  \  A )
64, 5eqtri 2496 . . 3  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  A )
72, 6uneq12i 3656 . 2  |-  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )  =  ( ( A  \  B )  u.  ( B  \  A ) )
81, 7eqtr2i 2497 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  ( ( A  u.  B )  \  ( A  i^i  B ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3473    u. cun 3474    i^i cin 3475
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-or 370  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-un 3481  df-in 3483
This theorem is referenced by: (None)
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