Theorem symdifv 5051

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 4207	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 3971	. . . . 5 ⊢ 𝐴 ⊆ V
3		ssdif0 4328	. . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 229	. . . 4 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4124	. . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		uncom 4118	. . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7		un0 4355	. . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
8	6, 7	eqtri 2759	. . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
9	5, 8	eqtri 2759	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
10	1, 9	eqtri 2759	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

Syntax hints:   = wceq 1541  Vcvv 3446   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913   △ csymdif 4206  ∅c0 4287

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-symdif 4207  df-nul 4288

This theorem is referenced by: (None)