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Theorem brsymdif 5208
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 5150 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆))
2 elsymdif 4248 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ ¬ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
3 df-br 5150 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4 df-br 5150 . . . 4 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
53, 4bibi12i 340 . . 3 ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
62, 5xchbinxr 335 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))
71, 6bitri 275 1 (𝐴(𝑅𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wcel 2107  csymdif 4242  cop 4635   class class class wbr 5149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-symdif 4243  df-br 5150
This theorem is referenced by:  brtxpsd  34866
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