Theorem elsymdif 4186

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 3893	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
3		eldif 3893	. . . 4 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵))
4	2, 3	orbi12i 920	. . 3 ⊢ ((𝐴 ∈ (𝐵 ∖ 𝐶) ∨ 𝐴 ∈ (𝐶 ∖ 𝐵)) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
5	1, 4	bitri 276	. 2 ⊢ (𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
6		df-symdif 4181	. . 3 ⊢ (𝐵 △ 𝐶) = ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵))
7	6	eleq2i 2831	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ 𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)))
8		xor 1022	. 2 ⊢ (¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) ↔ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) ∨ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵)))
9	5, 7, 8	3bitr4i 304	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∈ wcel 2119   ∖ cdif 3880   ∪ cun 3881   △ csymdif 4180

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-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-symdif 4181

This theorem is referenced by:  dfsymdif4  4187  elsymdifxor  4188  brsymdif  5131