Theorem elsymdif 4219

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 4115	. . 3 ⊢ (𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) ↔ (𝐴 ∈ (𝐵 ∖ 𝐶) ∨ 𝐴 ∈ (𝐶 ∖ 𝐵)))
2		eldif 3923	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
3		eldif 3923	. . . 4 ⊢ (𝐴 ∈ (𝐶 ∖ 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ ¬ 𝐴 ∈ 𝐵))
4	2, 3	orbi12i 927	. . 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 4214	. . 3 ⊢ (𝐵 △ 𝐶) = ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵))
7	6	eleq2i 2861	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ 𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)))
8		xor 1030	. 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))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∈ wcel 2149   ∖ cdif 3910   ∪ cun 3911   △ csymdif 4213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-symdif 4214

This theorem is referenced by:  dfsymdif4  4220  elsymdifxor  4221  brsymdif  5174