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Metamath Proof Explorer

Theorem brdif

Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011)

Ref Expression
Assertion brdif ⊢ A R ∖ S B ↔ A R B ∧ ¬ A S B

Proof

Step Hyp Ref Expression
1 eldif ⊢ A B ∈ R ∖ S ↔ A B ∈ R ∧ ¬ A B ∈ S
2 df-br ⊢ A R ∖ S B ↔ A B ∈ R ∖ S
3 df-br ⊢ A R B ↔ A B ∈ R
4 df-br ⊢ A S B ↔ A B ∈ S
5 4 notbii ⊢ ¬ A S B ↔ ¬ A B ∈ S
6 3 5 anbi12i ⊢ A R B ∧ ¬ A S B ↔ A B ∈ R ∧ ¬ A B ∈ S
7 1 2 6 3bitr4i ⊢ A R ∖ S B ↔ A R B ∧ ¬ A S B