Theorem neli 2406

Description: Inference associated with df-nel 2405. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
neli.1	⊢ 𝐴 ∉ 𝐵

Assertion

Ref	Expression
neli	⊢ ¬ 𝐴 ∈ 𝐵

Proof of Theorem neli

Step	Hyp	Ref	Expression
1		neli.1	. 2 ⊢ 𝐴 ∉ 𝐵
2		df-nel 2405	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	mpbi 144	1 ⊢ ¬ 𝐴 ∈ 𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 1481   ∉ wnel 2404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105

This theorem depends on definitions:  df-bi 116  df-nel 2405

This theorem is referenced by:  renepnf  7837  renemnf  7838  nn0nepnf  9072  xrltnr  9596  pnfnlt  9603  nltmnf  9604  inftonninf  10245  topnex  12294