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Theorem symdifid 23781
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 23773 . 2  |-  ( A(++) A )  =  ( ( A  \  A
)  u.  ( A 
\  A ) )
2 difid 3522 . . 3  |-  ( A 
\  A )  =  (/)
32, 2uneq12i 3327 . 2  |-  ( ( A  \  A )  u.  ( A  \  A ) )  =  ( (/)  u.  (/) )
4 un0 3479 . 2  |-  ( (/)  u.  (/) )  =  (/)
51, 3, 43eqtri 2307 1  |-  ( A(++) A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    \ cdif 3149    u. cun 3150   (/)c0 3455  (++)csymdif 23772
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-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-symdif 23773
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