Theorem eqneltri 2855

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 2827	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)
4	1, 3	mtbir 323	1 ⊢ ¬ 𝐴 ∈ 𝐶

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2728  df-clel 2811

This theorem is referenced by:  iprc  7862  tfr2b  8335  tz7.48-3  8383  pnfnre  11186  mnfnre  11188  prmrec  16893  00lsp  20976  nowisdomv  30544  goaln0  35575  bj-pinftynrr  37536  bj-minftynrr  37540  eliuniincex  45539  eliincex  45540  salgencntex  46771  nfermltl2rev  48219