Theorem symdif0 4533

Description: Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)

Assertion

Ref	Expression
symdif0	⊢ (𝐴 △ ∅) = 𝐴

Proof of Theorem symdif0

Step	Hyp	Ref	Expression
1		df-symdif 3806	. 2 ⊢ (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2		dif0 3904	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
3		0dif 3929	. . 3 ⊢ (∅ ∖ 𝐴) = ∅
4	2, 3	uneq12i 3727	. 2 ⊢ ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5		un0 3919	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
6	1, 4, 5	3eqtri 2636	1 ⊢ (𝐴 △ ∅) = 𝐴

Syntax hints:   = wceq 1475   ∖ cdif 3537   ∪ cun 3538   △ csymdif 3805  ∅c0 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875

This theorem is referenced by: (None)