Theorem symdifxor 3475

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 3210	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldif 3210	. . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	orbi12i 772	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
4		elun 3350	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)))
5		excxor 1423	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)))
6		ancom 266	. . . . 5 ⊢ ((¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
7	6	orbi2i 770	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (¬ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
8	5, 7	bitri 184	. . 3 ⊢ ((𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
9	3, 4, 8	3bitr4i 212	. 2 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵))
10	9	abbi2i 2346	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 716   = wceq 1398   ⊻ wxo 1420   ∈ wcel 2202  {cab 2217   ∖ cdif 3198   ∪ cun 3199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-xor 1421  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205

This theorem is referenced by: (None)