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Theorem symdif2 3814
 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 ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem symdif2
StepHypRef Expression
1 eldif 3550 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 eldif 3550 . . . 4 (𝑥 ∈ (𝐵𝐴) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐴))
31, 2orbi12i 542 . . 3 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
4 elun 3715 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐵𝐴)))
5 xor 931 . . 3 (¬ (𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∨ (𝑥𝐵 ∧ ¬ 𝑥𝐴)))
63, 4, 53bitr4i 291 . 2 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ ¬ (𝑥𝐴𝑥𝐵))
76abbi2i 2725 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ ¬ (𝑥𝐴𝑥𝐵)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ∖ cdif 3537   ∪ cun 3538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545 This theorem is referenced by:  mbfeqalem  23215
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