Theorem nelir 2458

Description: Inference associated with df-nel 2456. (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 2456	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
3	1, 2	mpbir 146	1 ⊢ 𝐴 ∉ 𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2160   ∉ wnel 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-nel 2456

This theorem is referenced by:  prneli  3632  snnex  4466  ruv  4567  pnfnre  8028  mnfnre  8029  eirr  11817  sqrt2irr  12193  topnex  14038