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Theorem symdifid 5090
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 4242 . 2 (𝐴𝐴) = ((𝐴𝐴) ∪ (𝐴𝐴))
2 difid 4370 . . 3 (𝐴𝐴) = ∅
32, 2uneq12i 4161 . 2 ((𝐴𝐴) ∪ (𝐴𝐴)) = (∅ ∪ ∅)
4 un0 4390 . 2 (∅ ∪ ∅) = ∅
51, 3, 43eqtri 2765 1 (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cdif 3945  cun 3946  csymdif 4241  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-symdif 4242  df-nul 4323
This theorem is referenced by: (None)
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