Theorem symdif0 5050

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 4201	. 2 ⊢ (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2		dif0 4330	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
3		0dif 4359	. . 3 ⊢ (∅ ∖ 𝐴) = ∅
4	2, 3	uneq12i 4168	. 2 ⊢ ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5		un0 4347	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
6	1, 4, 5	3eqtri 2790	1 ⊢ (𝐴 △ ∅) = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 4048   ∪ cun 4152   △ csymdif 4200  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-symdif 4201  df-nul 4282

This theorem is used by: (None)