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Theorem elsymdif 25660
Description: Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
elsymdif  |-  ( A  e.  ( B(++) C
)  <->  -.  ( A  e.  B  <->  A  e.  C
) )

Proof of Theorem elsymdif
StepHypRef Expression
1 elun 3480 . . 3  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( A  e.  ( B  \  C
)  \/  A  e.  ( C  \  B
) ) )
2 eldif 3322 . . . 4  |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
3 eldif 3322 . . . 4  |-  ( A  e.  ( C  \  B )  <->  ( A  e.  C  /\  -.  A  e.  B ) )
42, 3orbi12i 508 . . 3  |-  ( ( A  e.  ( B 
\  C )  \/  A  e.  ( C 
\  B ) )  <-> 
( ( A  e.  B  /\  -.  A  e.  C )  \/  ( A  e.  C  /\  -.  A  e.  B
) ) )
51, 4bitri 241 . 2  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
6 df-symdif 25655 . . 3  |-  ( B(++) C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
76eleq2i 2499 . 2  |-  ( A  e.  ( B(++) C
)  <->  A  e.  (
( B  \  C
)  u.  ( C 
\  B ) ) )
8 xor 862 . 2  |-  ( -.  ( A  e.  B  <->  A  e.  C )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
95, 7, 83bitr4i 269 1  |-  ( A  e.  ( B(++) C
)  <->  -.  ( A  e.  B  <->  A  e.  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    e. wcel 1725    \ cdif 3309    u. cun 3310  (++)csymdif 25654
This theorem is referenced by:  symdifass  25664  brsymdif  25665
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-un 3317  df-symdif 25655
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