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Theorem elsymdif 24367
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 3316 . . 3  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( A  e.  ( B  \  C
)  \/  A  e.  ( C  \  B
) ) )
2 eldif 3162 . . . 4  |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
3 eldif 3162 . . . 4  |-  ( A  e.  ( C  \  B )  <->  ( A  e.  C  /\  -.  A  e.  B ) )
42, 3orbi12i 507 . . 3  |-  ( ( A  e.  ( B 
\  C )  \/  A  e.  ( C 
\  B ) )  <-> 
( ( A  e.  B  /\  -.  A  e.  C )  \/  ( A  e.  C  /\  -.  A  e.  B
) ) )
51, 4bitri 240 . 2  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
6 df-symdif 24362 . . 3  |-  ( B(++) C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
76eleq2i 2347 . 2  |-  ( A  e.  ( B(++) C
)  <->  A  e.  (
( B  \  C
)  u.  ( C 
\  B ) ) )
8 xor 861 . 2  |-  ( -.  ( A  e.  B  <->  A  e.  C )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
95, 7, 83bitr4i 268 1  |-  ( A  e.  ( B(++) C
)  <->  -.  ( A  e.  B  <->  A  e.  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    e. wcel 1684    \ cdif 3149    u. cun 3150  (++)csymdif 24361
This theorem is referenced by:  symdifass  24371  brsymdif  24372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-un 3157  df-symdif 24362
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