Theorem nelir 2342

Description: Inference associated with df-nel 2340. (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 2340	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	mpbir 144	1 ⊢ 𝐴 ∉ 𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 1433   ∉ wnel 2339

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-nel 2340

This theorem is referenced by:  snnex  4199  ruv  4293  pnfnre  7160  mnfnre  7161  sqrt2irr  10541