Theorem symdif0 5096

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

Step	Hyp	Ref	Expression
1		df-symdif 4252	. 2 ⊢ (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2		dif0 4381	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
3		0dif 4410	. . 3 ⊢ (∅ ∖ 𝐴) = ∅
4	2, 3	uneq12i 4169	. 2 ⊢ ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5		un0 4399	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
6	1, 4, 5	3eqtri 2761	1 ⊢ (𝐴 △ ∅) = 𝐴

Syntax hints:   = wceq 1534   ∖ cdif 3955   ∪ cun 3956   △ csymdif 4251  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-rab 3429  df-v 3474  df-dif 3961  df-un 3963  df-ss 3975  df-symdif 4252  df-nul 4334

This theorem is referenced by: (None)