Theorem brsymdif

Description: Characterization of the symmetric difference of two binary relations. The RHS could also be expressed using exclusive disjunction df-xor . (Contributed by Scott Fenton, 11-Apr-2012)

	Ref	Expression
Assertion	brsymdif	$$⊢A\left(R∆S\right)B\leftrightarrow \neg \left(ARB\leftrightarrow ASB\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-br	$$⊢A\left(R∆S\right)B\leftrightarrow ⟨A,B⟩\in \left(R∆S\right)$$
2		elsymdif	$$⊢⟨A,B⟩\in \left(R∆S\right)\leftrightarrow \neg \left(⟨A,B⟩\in R\leftrightarrow ⟨A,B⟩\in S\right)$$
3		df-br	$$⊢ARB\leftrightarrow ⟨A,B⟩\in R$$
4		df-br	$$⊢ASB\leftrightarrow ⟨A,B⟩\in S$$
5	3 4	bibi12i	$$⊢\left(ARB\leftrightarrow ASB\right)\leftrightarrow \left(⟨A,B⟩\in R\leftrightarrow ⟨A,B⟩\in S\right)$$
6	2 5	xchbinxr	$$⊢⟨A,B⟩\in \left(R∆S\right)\leftrightarrow \neg \left(ARB\leftrightarrow ASB\right)$$
7	1 6	bitri	$$⊢A\left(R∆S\right)B\leftrightarrow \neg \left(ARB\leftrightarrow ASB\right)$$