Theorem symdif0 5010

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 4173	. 2 ⊢ (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2		dif0 4303	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
3		0dif 4332	. . 3 ⊢ (∅ ∖ 𝐴) = ∅
4	2, 3	uneq12i 4091	. 2 ⊢ ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5		un0 4321	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
6	1, 4, 5	3eqtri 2770	1 ⊢ (𝐴 △ ∅) = 𝐴

Syntax hints:   = wceq 1539   ∖ cdif 3880   ∪ cun 3881   △ csymdif 4172  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-symdif 4173  df-nul 4254

This theorem is referenced by: (None)