Theorem symdif2 3521

Description: Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem symdif2

Step	Hyp	Ref	Expression
1		eldif 3222	. . . 4 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
2		eldif 3222	. . . 4 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
3	1, 2	orbi12i 507	. . 3 ⊢ ((x ∈ (A ∖ B) ∨ x ∈ (B ∖ A)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∨ (x ∈ B ∧ ¬ x ∈ A)))
4		elun 3221	. . 3 ⊢ (x ∈ ((A ∖ B) ∪ (B ∖ A)) ↔ (x ∈ (A ∖ B) ∨ x ∈ (B ∖ A)))
5		xor 861	. . 3 ⊢ (¬ (x ∈ A ↔ x ∈ B) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∨ (x ∈ B ∧ ¬ x ∈ A)))
6	3, 4, 5	3bitr4i 268	. 2 ⊢ (x ∈ ((A ∖ B) ∪ (B ∖ A)) ↔ ¬ (x ∈ A ↔ x ∈ B))
7	6	abbi2i 2465	1 ⊢ ((A ∖ B) ∪ (B ∖ A)) = {x ∣ ¬ (x ∈ A ↔ x ∈ B)}

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   ∖ cdif 3207   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216

This theorem is referenced by: (None)