Theorem elsymdif 4181

Description: Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
elsymdif	⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))

Proof of Theorem elsymdif

Step	Hyp	Ref	Expression
1		elun 4083	. . 3 ⊢ (𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) ↔ (𝐴 ∈ (𝐵 ∖ 𝐶) ∨ 𝐴 ∈ (𝐶 ∖ 𝐵)))
2		eldif 3897	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
3		eldif 3897	. . . 4 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵))
4	2, 3	orbi12i 912	. . 3 ⊢ ((𝐴 ∈ (𝐵 ∖ 𝐶) ∨ 𝐴 ∈ (𝐶 ∖ 𝐵)) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
5	1, 4	bitri 274	. 2 ⊢ (𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
6		df-symdif 4176	. . 3 ⊢ (𝐵 △ 𝐶) = ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵))
7	6	eleq2i 2830	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ 𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)))
8		xor 1012	. 2 ⊢ (¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
9	5, 7, 8	3bitr4i 303	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∈ wcel 2106   ∖ cdif 3884   ∪ cun 3885   △ csymdif 4175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-symdif 4176

This theorem is referenced by:  dfsymdif4  4182  elsymdifxor  4183  brsymdif  5133