Theorem symdif0 5028

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 4198	. 2 ⊢ (𝐴 △ ∅) = ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴))
2		dif0 4323	. . 3 ⊢ (𝐴 ∖ ∅) = 𝐴
3		0dif 4350	. . 3 ⊢ (∅ ∖ 𝐴) = ∅
4	2, 3	uneq12i 4111	. 2 ⊢ ((𝐴 ∖ ∅) ∪ (∅ ∖ 𝐴)) = (𝐴 ∪ ∅)
5		un0 4339	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
6	1, 4, 5	3eqtri 2758	1 ⊢ (𝐴 △ ∅) = 𝐴

Syntax hints:   = wceq 1541   ∖ cdif 3894   ∪ cun 3895   △ csymdif 4197  ∅c0 4278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-symdif 4198  df-nul 4279

This theorem is referenced by: (None)