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Theorem symdifxor 3310
Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
Assertion
Ref Expression
symdifxor  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
Distinct variable groups:    x, A    x, B

Proof of Theorem symdifxor
StepHypRef Expression
1 eldif 3048 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3048 . . . 4  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
31, 2orbi12i 736 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( B 
\  A ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  B  /\  -.  x  e.  A
) ) )
4 elun 3185 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( B  \  A
) ) )
5 excxor 1339 . . . 4  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( -.  x  e.  A  /\  x  e.  B )
) )
6 ancom 264 . . . . 5  |-  ( ( -.  x  e.  A  /\  x  e.  B
)  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76orbi2i 734 . . . 4  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( -.  x  e.  A  /\  x  e.  B )
)  <->  ( ( x  e.  A  /\  -.  x  e.  B )  \/  ( x  e.  B  /\  -.  x  e.  A
) ) )
85, 7bitri 183 . . 3  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  B  /\  -.  x  e.  A )
) )
93, 4, 83bitr4i 211 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  A  \/_  x  e.  B ) )
109abbi2i 2230 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 680    = wceq 1314    \/_ wxo 1336    e. wcel 1463   {cab 2101    \ cdif 3036    u. cun 3037
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-xor 1337  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043
This theorem is referenced by: (None)
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