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Theorem brsymdif 4744
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 4686 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆))
2 elsymdif 3882 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ ¬ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
3 df-br 4686 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4 df-br 4686 . . . 4 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
53, 4bibi12i 328 . . 3 ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
62, 5xchbinxr 324 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))
71, 6bitri 264 1 (𝐴(𝑅𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 2030  csymdif 3876  cop 4216   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-symdif 3877  df-br 4686
This theorem is referenced by:  brtxpsd  32126
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