Theorem eqneltri 2888

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 2860	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)
4	1, 3	mtbir 326	1 ⊢ ¬ 𝐴 ∈ 𝐶

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-clel 2844

This theorem is referenced by:  iprc  7907  tfr2b  8382  tz7.48-3  8430  pnfnre  11249  mnfnre  11251  prmrec  16981  00lsp  21079  nowisdomv  30765  goaln0  35783  bj-pinftynrr  37753  bj-minftynrr  37757  eliuniincex  45718  eliincex  45719  salgencntex  46948  nfermltl2rev  48396