Theorem eqneltri 2879

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 2851	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)
4	1, 3	mtbir 326	1 ⊢ ¬ 𝐴 ∈ 𝐶

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-clel 2835

This theorem is used by:  iprc  7927  tfr2b  8411  tz7.48-3  8461  pnfnre  11357  mnfnre  11359  prmrec  17109  00lsp  21265  nowisdomv  31084  goaln0  36248  bj-pinftynrr  38252  bj-minftynrr  38256  eliuniincex  46232  eliincex  46233  salgencntex  47461  chnerlem1  48000  sqrtrrnpoly  48050  nfermltl2rev  48949