Theorem symdifxor 3497

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 3229	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldif 3229	. . . 4 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	orbi12i 776	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)))
4		elun 3370	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐵 ∖ 𝐴)))
5		excxor 1427	. . . 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 774	. . . 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 2353	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)}

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 720   = wceq 1402   ⊻ wxo 1424   ∈ wcel 2209  {cab 2224   ∖ cdif 3217   ∪ cun 3218

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-xor 1425  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224

This theorem is referenced by: (None)