Theorem eqneltri 2883

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 2855	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)
4	1, 3	mtbir 326	1 ⊢ ¬ 𝐴 ∈ 𝐶

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-clel 2839

This theorem is used by:  iprc  7957  tfr2b  8442  tz7.48-3  8492  pnfnre  11389  mnfnre  11391  prmrec  17144  00lsp  21307  nowisdomv  31132  goaln0  36366  bj-pinftynrr  38411  bj-minftynrr  38415  eliuniincex  46397  eliincex  46398  salgencntex  47626  chnerlem1  48165  sqrtrrnpoly  48215  nfermltl2rev  49114