Theorem brvvdif

Description: Binary relation with the complement under the universal class of ordered pairs. (Contributed by Peter Mazsa, 9-Nov-2018)

	Ref	Expression
Assertion	brvvdif	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ( V × V ) ∖ 𝑅 ) 𝐵 ↔ ¬ 𝐴 𝑅 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		opelvvdif	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ( V × V ) ∖ 𝑅 ) ↔ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ) )
2		df-br	⊢ ( 𝐴 ( ( V × V ) ∖ 𝑅 ) 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ( V × V ) ∖ 𝑅 ) )
3		df-br	⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
4	3	notbii	⊢ ( ¬ 𝐴 𝑅 𝐵 ↔ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
5	1 2 4	3bitr4g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ( ( V × V ) ∖ 𝑅 ) 𝐵 ↔ ¬ 𝐴 𝑅 𝐵 ) )