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Theorem eqneltri 2853
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 2825 . 2 (𝐴𝐶𝐵𝐶)
41, 3mtbir 323 1 ¬ 𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-clel 2811
This theorem is referenced by:  iprc  7904  tfr2b  8396  tz7.48-3  8444  pnfnre  11255  mnfnre  11257  prmrec  16855  00lsp  20592  goaln0  34384  bj-pinftynrr  36103  bj-minftynrr  36107  eliuniincex  43798  eliincex  43799  salgencntex  45059  nfermltl2rev  46411
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