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Theorem symdif0 4527
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 3805 . 2 (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2 dif0 3903 . . 3 (𝐴 ∖ ∅) = 𝐴
3 0dif 3928 . . 3 (∅ ∖ 𝐴) = ∅
42, 3uneq12i 3726 . 2 ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5 un0 3918 . 2 (𝐴 ∪ ∅) = 𝐴
61, 4, 53eqtri 2635 1 (𝐴 △ ∅) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  cdif 3536  cun 3537  csymdif 3804  c0 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-symdif 3805  df-nul 3874
This theorem is referenced by: (None)
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