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Theorem symdif1 3247
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 3235 . 2  |-  ( ( A  u.  B ) 
\  ( A  i^i  B ) )  =  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )
2 difin 3219 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
3 incom 3176 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43difeq2i 3099 . . . 4  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  ( B  i^i  A ) )
5 difin 3219 . . . 4  |-  ( B 
\  ( B  i^i  A ) )  =  ( B  \  A )
64, 5eqtri 2103 . . 3  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  A )
72, 6uneq12i 3136 . 2  |-  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )  =  ( ( A  \  B )  u.  ( B  \  A ) )
81, 7eqtr2i 2104 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  ( ( A  u.  B )  \  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    \ cdif 2981    u. cun 2982    i^i cin 2983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990
This theorem is referenced by: (None)
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