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 ) )