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 1570   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-clel 2835

This theorem is used by:  iprc  7908  tfr2b  8385  tz7.48-3  8433  pnfnre  11274  mnfnre  11276  prmrec  17014  00lsp  21165  nowisdomv  30954  goaln0  35972  bj-pinftynrr  37974  bj-minftynrr  37978  eliuniincex  45941  eliincex  45942  salgencntex  47171  chnerlem1  47710  sqrtrrnpoly  47760  nfermltl2rev  48659