Theorem symdifv 5088

Description: The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.)

Assertion

Ref	Expression
symdifv	⊢ (𝐴 △ V) = (V ∖ 𝐴)

Proof of Theorem symdifv

Step	Hyp	Ref	Expression
1		df-symdif 4241	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 4005	. . . . 5 ⊢ 𝐴 ⊆ V
3		ssdif0 4362	. . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 229	. . . 4 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4158	. . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		uncom 4152	. . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7		un0 4389	. . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
8	6, 7	eqtri 2758	. . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
9	5, 8	eqtri 2758	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
10	1, 9	eqtri 2758	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

Syntax hints:   = wceq 1539  Vcvv 3472   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947   △ csymdif 4240  ∅c0 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-symdif 4241  df-nul 4322

This theorem is referenced by: (None)