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Theorem symdif0 4731
Description: Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)
Assertion
Ref Expression
symdif0 (𝐴 △ ∅) = 𝐴

Proof of Theorem symdif0
StepHypRef Expression
1 df-symdif 3993 . 2 (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2 dif0 4097 . . 3 (𝐴 ∖ ∅) = 𝐴
3 0dif 4121 . . 3 (∅ ∖ 𝐴) = ∅
42, 3uneq12i 3916 . 2 ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5 un0 4111 . 2 (𝐴 ∪ ∅) = 𝐴
61, 4, 53eqtri 2797 1 (𝐴 △ ∅) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3720  cun 3721  csymdif 3992  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-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3993  df-nul 4064
This theorem is referenced by: (None)
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