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Theorem symdifid 4977
 Description: The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)
Assertion
Ref Expression
symdifid (𝐴𝐴) = ∅

Proof of Theorem symdifid
StepHypRef Expression
1 df-symdif 4149 . 2 (𝐴𝐴) = ((𝐴𝐴) ∪ (𝐴𝐴))
2 difid 4271 . . 3 (𝐴𝐴) = ∅
32, 2uneq12i 4068 . 2 ((𝐴𝐴) ∪ (𝐴𝐴)) = (∅ ∪ ∅)
4 un0 4289 . 2 (∅ ∪ ∅) = ∅
51, 3, 43eqtri 2785 1 (𝐴𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∖ cdif 3857   ∪ cun 3858   △ csymdif 4148  ∅c0 4227 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-symdif 4149  df-nul 4228 This theorem is referenced by: (None)
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