Metamath Proof Explorer


Theorem brdif

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

Ref Expression
Assertion brdif ( 𝐴 ( 𝑅𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ ¬ 𝐴 𝑆 𝐵 ) )

Proof

Step Hyp Ref Expression
1 eldif ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅𝑆 ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ∧ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑆 ) )
2 df-br ( 𝐴 ( 𝑅𝑆 ) 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( 𝑅𝑆 ) )
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 ( 𝑅𝑆 ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∧ ¬ 𝐴 𝑆 𝐵 ) )