Theorem elsymdif 4211

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 4107	. . 3 ⊢ (𝐴 ∈ ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵)) ↔ (𝐴 ∈ (𝐵 ∖ 𝐶) ∨ 𝐴 ∈ (𝐶 ∖ 𝐵)))
2		eldif 3915	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))
3		eldif 3915	. . . 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 4206	. . 3 ⊢ (𝐵 △ 𝐶) = ((𝐵 ∖ 𝐶) ∪ (𝐶 ∖ 𝐵))
7	6	eleq2i 2855	. 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))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∈ wcel 2143   ∖ cdif 3902   ∪ cun 3903   △ csymdif 4205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908  df-un 3910  df-symdif 4206

This theorem is referenced by:  dfsymdif4  4212  elsymdifxor  4213  brsymdif  5170