Theorem brsymdif 5159

Description: Characterization of the symmetric difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2012.)

Assertion

Ref	Expression
brsymdif	⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))

Proof of Theorem brsymdif

Step	Hyp	Ref	Expression
1		df-br 5101	. 2 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆))
2		elsymdif 4212	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
3		df-br 5101	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5101	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	bibi12i 339	. . 3 ⊢ ((𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	2, 5	xchbinxr 335	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
7	1, 6	bitri 275	1 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∈ wcel 2114   △ csymdif 4206  ⟨cop 4588   class class class wbr 5100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-un 3908  df-symdif 4207  df-br 5101

This theorem is referenced by:  brtxpsd  36108