Theorem neli 2464

Description: Inference associated with df-nel 2463. (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 2463	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	mpbi 145	1 ⊢ ¬ 𝐴 ∈ 𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2167   ∉ wnel 2462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem depends on definitions:  df-bi 117  df-nel 2463

This theorem is referenced by:  renepnf  8074  renemnf  8075  nn0nepnf  9320  xrltnr  9854  pnfnlt  9862  nltmnf  9863  inftonninf  10534  pcgcd1  12497  pc2dvds  12499  topnex  14322