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Theorem symdifid 5080
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 4234 . 2 (𝐴𝐴) = ((𝐴𝐴) ∪ (𝐴𝐴))
2 difid 4362 . . 3 (𝐴𝐴) = ∅
32, 2uneq12i 4153 . 2 ((𝐴𝐴) ∪ (𝐴𝐴)) = (∅ ∪ ∅)
4 un0 4382 . 2 (∅ ∪ ∅) = ∅
51, 3, 43eqtri 2756 1 (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cdif 3937  cun 3938  csymdif 4233  c0 4314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-symdif 4234  df-nul 4315
This theorem is referenced by: (None)
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