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Theorem symdif0 4782
Description: Symmetric difference with the empty class. The empty class is the identity element for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdif0 (𝐴 △ ∅) = 𝐴

Proof of Theorem symdif0
StepHypRef Expression
1 df-symdif 4036 . 2 (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2 dif0 4145 . . 3 (𝐴 ∖ ∅) = 𝐴
3 0dif 4169 . . 3 (∅ ∖ 𝐴) = ∅
42, 3uneq12i 3958 . 2 ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5 un0 4159 . 2 (𝐴 ∪ ∅) = 𝐴
61, 4, 53eqtri 2828 1 (𝐴 △ ∅) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  cdif 3760  cun 3761  csymdif 4035  c0 4110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ral 3097  df-rab 3101  df-v 3389  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-symdif 4036  df-nul 4111
This theorem is referenced by: (None)
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