Theorem nelbrnelim

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

	Ref	Expression
Assertion	nelbrnelim	⊢ ( 𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nelbrim	⊢ ( 𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵 )
2		df-nel	⊢ ( 𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵 )
3	1 2	sylibr	⊢ ( 𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵 )