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Theorem symdif1 3593
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 3581 . 2  |-  ( ( A  u.  B ) 
\  ( A  i^i  B ) )  =  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )
2 difin 3565 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
3 incom 3520 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43difeq2i 3449 . . . 4  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  ( B  i^i  A ) )
5 difin 3565 . . . 4  |-  ( B 
\  ( B  i^i  A ) )  =  ( B  \  A )
64, 5eqtri 2450 . . 3  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  A )
72, 6uneq12i 3486 . 2  |-  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )  =  ( ( A  \  B )  u.  ( B  \  A ) )
81, 7eqtr2i 2451 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  ( ( A  u.  B )  \  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3304    u. cun 3305    i^i cin 3306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ral 2697  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314
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