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Theorem symdif0 4998
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 4216 . 2 (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2 dif0 4329 . . 3 (𝐴 ∖ ∅) = 𝐴
3 0dif 4352 . . 3 (∅ ∖ 𝐴) = ∅
42, 3uneq12i 4134 . 2 ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5 un0 4341 . 2 (𝐴 ∪ ∅) = 𝐴
61, 4, 53eqtri 2845 1 (𝐴 △ ∅) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  cdif 3930  cun 3931  csymdif 4215  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-symdif 4216  df-nul 4289
This theorem is referenced by: (None)
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