Theorem neli 2517

Description: Inference associated with df-nel 2516. (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 2516	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	mpbi 145	1 ⊢ ¬ 𝐴 ∈ 𝐵

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2209   ∉ wnel 2515

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This proof depends on definitions:  df-bi 117  df-nel 2516

This theorem is used by:  renepnf  8374  renemnf  8375  nn0nepnf  9643  xrltnr  10192  pnfnlt  10200  nltmnf  10201  inftonninf  10893  pcgcd1  13129  pc2dvds  13131  topnex  15239