Theorem symdifxor 3429

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 3166	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldif 3166	. . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	orbi12i 765	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
4		elun 3304	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)))
5		excxor 1389	. . . 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 763	. . . 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 2311	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 709   = wceq 1364   ⊻ wxo 1386   ∈ wcel 2167  {cab 2182   ∖ cdif 3154   ∪ cun 3155

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-xor 1387  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161

This theorem is referenced by: (None)