Theorem brsymdif 5125

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 5067	. 2 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆))
2		elsymdif 4224	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
3		df-br 5067	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5067	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	bibi12i 342	. . 3 ⊢ ((𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	2, 5	xchbinxr 337	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
7	1, 6	bitri 277	1 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∈ wcel 2114   △ csymdif 4218  ⟨cop 4573   class class class wbr 5066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-symdif 4219  df-br 5067

This theorem is referenced by:  brtxpsd  33355