Theorem symdifv 5054

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 4206	. 2 ⊢ (𝐴 △ V) = ((𝐴 ∖ V) ∪ (V ∖ 𝐴))
2		ssv 3962	. . . 4 ⊢ 𝐴 ⊆ V
3		ssdif0 4321	. . . 4 ⊢ (𝐴 ⊆ V ↔ (𝐴 ∖ V) = ∅)
4	2, 3	mpbi 233	. . 3 ⊢ (𝐴 ∖ V) = ∅
5	4	uneq1i 4118	. 2 ⊢ ((𝐴 ∖ V) ∪ (V ∖ 𝐴)) = (∅ ∪ (V ∖ 𝐴))
6		0un 4353	. 2 ⊢ (∅ ∪ (V ∖ 𝐴)) = (V ∖ 𝐴)
7	1, 5, 6	3eqtri 2792	1 ⊢ (𝐴 △ V) = (V ∖ 𝐴)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906   △ csymdif 4205  ∅c0 4286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-symdif 4206  df-nul 4287

This theorem is used by: (None)