Theorem elsymdif 4202

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 4151	. . 3 ⊢ (𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) ↔ (𝐴 ∈ (𝐵 ∖ 𝐶) ∨ 𝐴 ∈ (𝐶 ∖ 𝐵)))
2		eldif 4048	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
3		eldif 4048	. . . 4 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵))
4	2, 3	orbi12i 928	. . 3 ⊢ ((𝐴 ∈ (𝐵 ∖ 𝐶) ∨ 𝐴 ∈ (𝐶 ∖ 𝐵)) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
5	1, 4	bitri 278	. 2 ⊢ (𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
6		df-symdif 4197	. . 3 ⊢ (𝐵 △ 𝐶) = ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵))
7	6	eleq2i 2852	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ 𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)))
8		xor 1032	. 2 ⊢ (¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
9	5, 7, 8	3bitr4i 306	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∈ wcel 2144   ∖ cdif 4044   ∪ cun 4148   △ csymdif 4196

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4046  df-un 4150  df-symdif 4197

This theorem is used by:  dfsymdif4  4203  elsymdifxor  4204  brsymdif  5158