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Theorem symdif2 3447
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 3175 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3175 . . . 4  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
31, 2orbi12i 507 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( B 
\  A ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  B  /\  -.  x  e.  A
) ) )
4 elun 3329 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( B  \  A
) ) )
5 xor 861 . . 3  |-  ( -.  ( x  e.  A  <->  x  e.  B )  <->  ( (
x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  B  /\  -.  x  e.  A )
) )
63, 4, 53bitr4i 268 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  -.  (
x  e.  A  <->  x  e.  B ) )
76abbi2i 2407 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  -.  ( x  e.  A  <->  x  e.  B ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282    \ cdif 3162    u. cun 3163
This theorem is referenced by:  mbfeqalem  19013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-un 3170
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