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Theorem symdifid 25671
Description: Symmetric difference yields the empty class with the same argument twice. (Contributed by Scott Fenton, 25-Apr-2012.)
Assertion
Ref Expression
symdifid  |-  ( A(++) A )  =  (/)

Proof of Theorem symdifid
StepHypRef Expression
1 df-symdif 25663 . 2  |-  ( A(++) A )  =  ( ( A  \  A
)  u.  ( A 
\  A ) )
2 difid 3696 . . 3  |-  ( A 
\  A )  =  (/)
32, 2uneq12i 3499 . 2  |-  ( ( A  \  A )  u.  ( A  \  A ) )  =  ( (/)  u.  (/) )
4 un0 3652 . 2  |-  ( (/)  u.  (/) )  =  (/)
51, 3, 43eqtri 2460 1  |-  ( A(++) A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3317    u. cun 3318   (/)c0 3628  (++)csymdif 25662
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-symdif 25663
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