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Theorem brsymdif 4984
 Description: Characterization of the symmetric difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2012.)
Assertion
Ref Expression
brsymdif (𝐴(𝑅𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))

Proof of Theorem brsymdif
StepHypRef Expression
1 df-br 4926 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆))
2 elsymdif 4105 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ ¬ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
3 df-br 4926 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4 df-br 4926 . . . 4 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
53, 4bibi12i 332 . . 3 ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
62, 5xchbinxr 327 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))
71, 6bitri 267 1 (𝐴(𝑅𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   ∈ wcel 2051   △ csymdif 4099  ⟨cop 4441   class class class wbr 4925 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-v 3410  df-dif 3825  df-un 3827  df-symdif 4100  df-br 4926 This theorem is referenced by:  brtxpsd  32913
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