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Metamath Proof Explorer

Theorem brsymdif

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

Ref Expression
Assertion brsymdif ⊢ A R ∆ S B ↔ ¬ A R B ↔ A S B

Proof

Step Hyp Ref Expression
1 df-br ⊢ A R ∆ S B ↔ A B ∈ R ∆ S
2 elsymdif ⊢ A B ∈ R ∆ S ↔ ¬ A B ∈ R ↔ A B ∈ S
3 df-br ⊢ A R B ↔ A B ∈ R
4 df-br ⊢ A S B ↔ A B ∈ S
5 3 4 bibi12i ⊢ A R B ↔ A S B ↔ A B ∈ R ↔ A B ∈ S
6 2 5 xchbinxr ⊢ A B ∈ R ∆ S ↔ ¬ A R B ↔ A S B
7 1 6 bitri ⊢ A R ∆ S B ↔ ¬ A R B ↔ A S B