Theorem brsymdif 5172

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 5112	. 2 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆))
2		elsymdif 4211	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
3		df-br 5112	. . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5112	. . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	bibi12i 342	. . 3 ⊢ ((𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	2, 5	xchbinxr 338	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
7	1, 6	bitri 278	1 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∈ wcel 2146   △ csymdif 4205  ⟨cop 4597   class class class wbr 5111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3909  df-un 3911  df-symdif 4206  df-br 5112

This theorem is used by:  brtxpsd  36423