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Theorem symdif1 3311
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 3299 . 2  |-  ( ( A  u.  B ) 
\  ( A  i^i  B ) )  =  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )
2 difin 3283 . . 3  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
3 incom 3238 . . . . 5  |-  ( A  i^i  B )  =  ( B  i^i  A
)
43difeq2i 3161 . . . 4  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  ( B  i^i  A ) )
5 difin 3283 . . . 4  |-  ( B 
\  ( B  i^i  A ) )  =  ( B  \  A )
64, 5eqtri 2138 . . 3  |-  ( B 
\  ( A  i^i  B ) )  =  ( B  \  A )
72, 6uneq12i 3198 . 2  |-  ( ( A  \  ( A  i^i  B ) )  u.  ( B  \ 
( A  i^i  B
) ) )  =  ( ( A  \  B )  u.  ( B  \  A ) )
81, 7eqtr2i 2139 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  ( ( A  u.  B )  \  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1316    \ cdif 3038    u. cun 3039    i^i cin 3040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047
This theorem is referenced by: (None)
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