Theorem symdifv 5032

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 4200	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 3954	. . . . 5 ⊢ 𝐴 ⊆ V
3		ssdif0 4313	. . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 230	. . . 4 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4111	. . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		uncom 4105	. . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7		un0 4341	. . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
8	6, 7	eqtri 2754	. . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
9	5, 8	eqtri 2754	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
10	1, 9	eqtri 2754	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

Syntax hints:   = wceq 1541  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ⊆ wss 3897   △ csymdif 4199  ∅c0 4280

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-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-symdif 4200  df-nul 4281

This theorem is referenced by: (None)