Theorem symdifxor 3197

Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)

Assertion

Ref	Expression
symdifxor	⊢ ((A ∖ B) ∪ (B ∖ A)) = {x ∣ (x ∈ A ⊻ x ∈ B)}

Distinct variable groups:   x,A   x,B

Proof of Theorem symdifxor

Step	Hyp	Ref	Expression
1		eldif 2921	. . . 4 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
2		eldif 2921	. . . 4 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A))
3	1, 2	orbi12i 680	. . 3 ⊢ ((x ∈ (A ∖ B) ∨ x ∈ (B ∖ A)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∨ (x ∈ B ∧ ¬ x ∈ A)))
4		elun 3078	. . 3 ⊢ (x ∈ ((A ∖ B) ∪ (B ∖ A)) ↔ (x ∈ (A ∖ B) ∨ x ∈ (B ∖ A)))
5		excxor 1268	. . . 4 ⊢ ((x ∈ A ⊻ x ∈ B) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∨ (¬ x ∈ A ∧ x ∈ B)))
6		ancom 253	. . . . 5 ⊢ ((¬ x ∈ A ∧ x ∈ B) ↔ (x ∈ B ∧ ¬ x ∈ A))
7	6	orbi2i 678	. . . 4 ⊢ (((x ∈ A ∧ ¬ x ∈ B) ∨ (¬ x ∈ A ∧ x ∈ B)) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∨ (x ∈ B ∧ ¬ x ∈ A)))
8	5, 7	bitri 173	. . 3 ⊢ ((x ∈ A ⊻ x ∈ B) ↔ ((x ∈ A ∧ ¬ x ∈ B) ∨ (x ∈ B ∧ ¬ x ∈ A)))
9	3, 4, 8	3bitr4i 201	. 2 ⊢ (x ∈ ((A ∖ B) ∪ (B ∖ A)) ↔ (x ∈ A ⊻ x ∈ B))
10	9	abbi2i 2149	1 ⊢ ((A ∖ B) ∪ (B ∖ A)) = {x ∣ (x ∈ A ⊻ x ∈ B)}

Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 628   = wceq 1242   ⊻ wxo 1265   ∈ wcel 1390  {cab 2023   ∖ cdif 2908   ∪ cun 2909

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-xor 1266  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916

This theorem is referenced by: (None)