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Theorem symdif1 3433
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 3422 . 2  |-  ( ( A  u.  B ) 
\  ( A  i^i  B ) )  =  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )
2 difin 3406 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
3 incom 3361 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43difeq2i 3291 . . . 4  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  ( B  i^i  A ) )
5 difin 3406 . . . 4  |-  ( B 
\  ( B  i^i  A ) )  =  ( B  \  A )
64, 5eqtri 2303 . . 3  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  A )
72, 6uneq12i 3327 . 2  |-  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )  =  ( ( A  \  B )  u.  ( B  \  A ) )
81, 7eqtr2i 2304 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  ( ( A  u.  B )  \  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    \ cdif 3149    u. cun 3150    i^i cin 3151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159
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