Theorem symdifv 5015

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 4181	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 3939	. . . 4 ⊢ 𝐴 ⊆ V
3		ssdif0 4294	. . . 4 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 231	. . 3 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4094	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		0un 4324	. 2 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
7	1, 5, 6	3eqtri 2766	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

Syntax hints:   = wceq 1547  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883   △ csymdif 4180  ∅c0 4261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-symdif 4181  df-nul 4262

This theorem is referenced by: (None)