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Theorem eqneltri 2852
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 ¬ 𝐵𝐶
Assertion
Ref Expression
eqneltri ¬ 𝐴𝐶

Proof of Theorem eqneltri
StepHypRef Expression
1 eqneltri.2 . 2 ¬ 𝐵𝐶
2 eqneltri.1 . . 3 𝐴 = 𝐵
32eleq1i 2824 . 2 (𝐴𝐶𝐵𝐶)
41, 3mtbir 322 1 ¬ 𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724  df-clel 2810
This theorem is referenced by:  iprc  7906  tfr2b  8398  tz7.48-3  8446  pnfnre  11257  mnfnre  11259  prmrec  16857  00lsp  20597  goaln0  34453  bj-pinftynrr  36189  bj-minftynrr  36193  eliuniincex  43880  eliincex  43881  salgencntex  45138  nfermltl2rev  46490
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