Theorem nelbrnel

Description: A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021)

	Ref	Expression
Assertion	nelbrnel	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 _∉ 𝐵 ↔ 𝐴 ∉ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nelbr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵 ) )
2		df-nel	⊢ ( 𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵 )
3	1 2	bitr4di	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 _∉ 𝐵 ↔ 𝐴 ∉ 𝐵 ) )