Theorem symdif2 1690

Description: Two ways of expressing symmetric difference.

Assertion

Ref	Expression
symdif2	⊢ ((A ∖ B) ∪ (B ∖ A)) = {x∣ ¬ (x ∈ A ↔ x ∈ B)}

Distinct variable group(s):   x,A   x,B

Proof of Theorem symdif2

Step	Hyp	Ref	Expression
1		elun 1601	. . 3 ⊢ (x ∈ ((A ∖ B) ∪ (B ∖ A)) ↔ (x ∈ (A ∖ B) ∨ x ∈ (B ∖ A)))
2		eldif 1496	. . . . 5 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
3		pm4.13 142	. . . . . 6 ⊢ (x ∈ A ↔ ¬ ¬ x ∈ A)
4	3	anbi1i 368	. . . . 5 ⊢ ((x ∈ A ∧ ¬ x ∈ B) ↔ (¬ ¬ x ∈ A ∧ ¬ x ∈ B))
5	2, 4	bitr 151	. . . 4 ⊢ (x ∈ (A ∖ B) ↔ (¬ ¬ x ∈ A ∧ ¬ x ∈ B))
6		eldif 1496	. . . . 5 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
7		ancom 333	. . . . 5 ⊢ ((x ∈ B ∧ ¬ x ∈ A) ↔ (¬ x ∈ A ∧ x ∈ B))
8	6, 7	bitr 151	. . . 4 ⊢ (x ∈ (B ∖ A) ↔ (¬ x ∈ A ∧ x ∈ B))
9	5, 8	orbi12i 216	. . 3 ⊢ ((x ∈ (A ∖ B) ∨ x ∈ (B ∖ A)) ↔ ((¬ ¬ x ∈ A ∧ ¬ x ∈ B) ∨ (¬ x ∈ A ∧ x ∈ B)))
10		orcom 209	. . . . 5 ⊢ (((¬ ¬ x ∈ A ∧ ¬ x ∈ B) ∨ (¬ x ∈ A ∧ x ∈ B)) ↔ ((¬ x ∈ A ∧ x ∈ B) ∨ (¬ ¬ x ∈ A ∧ ¬ x ∈ B)))
11		dfbi 499	. . . . 5 ⊢ ((¬ x ∈ A ↔ x ∈ B) ↔ ((¬ x ∈ A ∧ x ∈ B) ∨ (¬ ¬ x ∈ A ∧ ¬ x ∈ B)))
12	10, 11	bitr4 154	. . . 4 ⊢ (((¬ ¬ x ∈ A ∧ ¬ x ∈ B) ∨ (¬ x ∈ A ∧ x ∈ B)) ↔ (¬ x ∈ A ↔ x ∈ B))
13		nbbn 498	. . . 4 ⊢ ((¬ x ∈ A ↔ x ∈ B) ↔ ¬ (x ∈ A ↔ x ∈ B))
14	12, 13	bitr 151	. . 3 ⊢ (((¬ ¬ x ∈ A ∧ ¬ x ∈ B) ∨ (¬ x ∈ A ∧ x ∈ B)) ↔ ¬ (x ∈ A ↔ x ∈ B))
15	1, 9, 14	3bitr 155	. 2 ⊢ (x ∈ ((A ∖ B) ∪ (B ∖ A)) ↔ ¬ (x ∈ A ↔ x ∈ B))
16	15	biabri 1180	1 ⊢ ((A ∖ B) ∪ (B ∖ A)) = {x∣ ¬ (x ∈ A ↔ x ∈ B)}

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∪ cun 1485

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490

metamath.org