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Theorem symdif2 3607
Description: Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
symdif2  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  -.  ( x  e.  A  <->  x  e.  B ) }
Distinct variable groups:    x, A    x, B

Proof of Theorem symdif2
StepHypRef Expression
1 eldif 3330 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3330 . . . 4  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
31, 2orbi12i 508 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( B 
\  A ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  B  /\  -.  x  e.  A
) ) )
4 elun 3488 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( B  \  A
) ) )
5 xor 862 . . 3  |-  ( -.  ( x  e.  A  <->  x  e.  B )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  B  /\  -.  x  e.  A )
) )
63, 4, 53bitr4i 269 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  -.  (
x  e.  A  <->  x  e.  B ) )
76abbi2i 2547 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  -.  ( x  e.  A  <->  x  e.  B ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422    \ cdif 3317    u. cun 3318
This theorem is referenced by:  mbfeqalem  19534
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-un 3325
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