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Theorem elsymdif 3730
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 3641 . . 3  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( A  e.  ( B  \  C
)  \/  A  e.  ( C  \  B
) ) )
2 eldif 3481 . . . 4  |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
3 eldif 3481 . . . 4  |-  ( A  e.  ( C  \  B )  <->  ( A  e.  C  /\  -.  A  e.  B ) )
42, 3orbi12i 521 . . 3  |-  ( ( A  e.  ( B 
\  C )  \/  A  e.  ( C 
\  B ) )  <-> 
( ( A  e.  B  /\  -.  A  e.  C )  \/  ( A  e.  C  /\  -.  A  e.  B
) ) )
51, 4bitri 249 . 2  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
6 df-symdif 3725 . . 3  |-  ( B  /_\  C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
76eleq2i 2535 . 2  |-  ( A  e.  ( B  /_\  C )  <-> 
A  e.  ( ( B  \  C )  u.  ( C  \  B ) ) )
8 xor 891 . 2  |-  ( -.  ( A  e.  B  <->  A  e.  C )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
95, 7, 83bitr4i 277 1  |-  ( A  e.  ( B  /_\  C )  <->  -.  ( A  e.  B  <->  A  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    e. wcel 1819    \ cdif 3468    u. cun 3469    /_\ csymdif 3724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3474  df-un 3476  df-symdif 3725
This theorem is referenced by:  elsymdifxor  3731  symdifass  3734  brsymdif  4513
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