Theorem nelir 2518

Description: Inference associated with df-nel 2516. (Contributed by BJ, 7-Jul-2018.)

Hypothesis

Ref	Expression
nelir.1	⊢ ¬ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
nelir	⊢ 𝐴 ∉ 𝐵

Proof of Theorem nelir

Step	Hyp	Ref	Expression
1		nelir.1	. 2 ⊢ ¬ 𝐴 ∈ 𝐵
2		df-nel 2516	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	mpbir 146	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  ax-ia2 107  ax-ia3 108

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

This theorem is used by:  prneli  3734  snnex  4594  ruv  4697  pnfnre  8367  mnfnre  8368  eirr  12546  sqrt2irr  12940  topnex  15187