Theorem symdifv 5011

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 4222	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 3994	. . . . 5 ⊢ 𝐴 ⊆ V
3		ssdif0 4326	. . . . 5 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 232	. . . 4 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4138	. . 3 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		uncom 4132	. . . 4 ⊢ (∅ ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ ∅)
7		un0 4347	. . . 4 ⊢ ((V ∖ 𝐴) ∪ ∅) = (V ∖ 𝐴)
8	6, 7	eqtri 2847	. . 3 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
9	5, 8	eqtri 2847	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
10	1, 9	eqtri 2847	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

Syntax hints:   = wceq 1536  Vcvv 3497   ∖ cdif 3936   ∪ cun 3937   ⊆ wss 3939   △ csymdif 4221  ∅c0 4294

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295

This theorem is referenced by: (None)