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Theorem symdifid 29053
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 29045 . 2  |-  ( A(++) A )  =  ( ( A  \  A
)  u.  ( A 
\  A ) )
2 difid 3895 . . 3  |-  ( A 
\  A )  =  (/)
32, 2uneq12i 3656 . 2  |-  ( ( A  \  A )  u.  ( A  \  A ) )  =  ( (/)  u.  (/) )
4 un0 3810 . 2  |-  ( (/)  u.  (/) )  =  (/)
51, 3, 43eqtri 2500 1  |-  ( A(++) A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3473    u. cun 3474   (/)c0 3785  (++)csymdif 29044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-symdif 29045
This theorem is referenced by: (None)
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