Theorem symdifv 4971

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 4169	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 3939	. . . . 5 ⊢ 𝐴 ⊆ V
3		ssdif0 4277	. . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 233	. . . 4 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4086	. . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		uncom 4080	. . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7		un0 4298	. . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
8	6, 7	eqtri 2821	. . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
9	5, 8	eqtri 2821	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
10	1, 9	eqtri 2821	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

Syntax hints:   = wceq 1538  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ⊆ wss 3881   △ csymdif 4168  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-symdif 4169  df-nul 4244

This theorem is referenced by: (None)