Theorem symdifv 5043

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 4205	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 3960	. . . 4 ⊢ 𝐴 ⊆ V
3		ssdif0 4319	. . . 4 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 232	. . 3 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4117	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		0un 4350	. 2 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
7	1, 5, 6	3eqtri 2789	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

Syntax hints:   = wceq 1560  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904   △ csymdif 4204  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-symdif 4205  df-nul 4286

This theorem is referenced by: (None)