Theorem nelir 2353

Description: Inference associated with df-nel 2351. (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 2351	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	mpbir 144	1 ⊢ 𝐴 ∉ 𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 1438   ∉ wnel 2350

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 2351

This theorem is referenced by:  prneli  3466  snnex  4262  ruv  4356  pnfnre  7508  mnfnre  7509  sqrt2irr  11234