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Theorem eqneltri 2858
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 2830 . 2 (𝐴𝐶𝐵𝐶)
41, 3mtbir 323 1 ¬ 𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-clel 2814
This theorem is referenced by:  iprc  7934  tfr2b  8435  tz7.48-3  8483  pnfnre  11300  mnfnre  11302  prmrec  16956  00lsp  20997  goaln0  35378  bj-pinftynrr  37205  bj-minftynrr  37209  eliuniincex  45049  eliincex  45050  salgencntex  46299  nfermltl2rev  47668
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