Theorem eqneltri 2851

Description: If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
eqneltri.1	⊢ 𝐴 = 𝐵
eqneltri.2	⊢ ¬ 𝐵 ∈ 𝐶

Assertion

Ref	Expression
eqneltri	⊢ ¬ 𝐴 ∈ 𝐶

Proof of Theorem eqneltri

Step	Hyp	Ref	Expression
1		eqneltri.2	. 2 ⊢ ¬ 𝐵 ∈ 𝐶
2		eqneltri.1	. . 3 ⊢ 𝐴 = 𝐵
3	2	eleq1i 2823	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)
4	1, 3	mtbir 322	1 ⊢ ¬ 𝐴 ∈ 𝐶

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2723  df-clel 2809

This theorem is referenced by:  iprc  7855  tfr2b  8347  tz7.48-3  8395  pnfnre  11205  mnfnre  11207  prmrec  16805  00lsp  20499  goaln0  34074  bj-pinftynrr  35766  bj-minftynrr  35770  eliuniincex  43441  eliincex  43442  salgencntex  44704  nfermltl2rev  46055