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Theorem elsymdif 3698
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 3606 . . 3  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( A  e.  ( B  \  C
)  \/  A  e.  ( C  \  B
) ) )
2 eldif 3446 . . . 4  |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
3 eldif 3446 . . . 4  |-  ( A  e.  ( C  \  B )  <->  ( A  e.  C  /\  -.  A  e.  B ) )
42, 3orbi12i 523 . . 3  |-  ( ( A  e.  ( B 
\  C )  \/  A  e.  ( C 
\  B ) )  <-> 
( ( A  e.  B  /\  -.  A  e.  C )  \/  ( A  e.  C  /\  -.  A  e.  B
) ) )
51, 4bitri 252 . 2  |-  ( A  e.  ( ( B 
\  C )  u.  ( C  \  B
) )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
6 df-symdif 3693 . . 3  |-  ( B  /_\  C )  =  ( ( B  \  C
)  u.  ( C 
\  B ) )
76eleq2i 2499 . 2  |-  ( A  e.  ( B  /_\  C )  <-> 
A  e.  ( ( B  \  C )  u.  ( C  \  B ) ) )
8 xor 899 . 2  |-  ( -.  ( A  e.  B  <->  A  e.  C )  <->  ( ( A  e.  B  /\  -.  A  e.  C
)  \/  ( A  e.  C  /\  -.  A  e.  B )
) )
95, 7, 83bitr4i 280 1  |-  ( A  e.  ( B  /_\  C )  <->  -.  ( A  e.  B  <->  A  e.  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    \/ wo 369    /\ wa 370    e. wcel 1872    \ cdif 3433    u. cun 3434    /_\ csymdif 3692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-dif 3439  df-un 3441  df-symdif 3693
This theorem is referenced by:  elsymdifxor  3699  symdifass  3702  brsymdif  4480
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