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Theorem symdifid 4734
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 3994 . 2 (𝐴𝐴) = ((𝐴𝐴) ∪ (𝐴𝐴))
2 difid 4096 . . 3 (𝐴𝐴) = ∅
32, 2uneq12i 3916 . 2 ((𝐴𝐴) ∪ (𝐴𝐴)) = (∅ ∪ ∅)
4 un0 4112 . 2 (∅ ∪ ∅) = ∅
51, 3, 43eqtri 2797 1 (𝐴𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3720  cun 3721  csymdif 3993  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3994  df-nul 4064
This theorem is referenced by: (None)
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