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Theorem eqneltri 2909
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 2906 . 2 (𝐴𝐶𝐵𝐶)
41, 3mtbir 326 1 ¬ 𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2817  df-clel 2896
This theorem is referenced by:  iprc  7610  tfr2b  8024  tz7.48-3  8072  pnfnre  10676  mnfnre  10678  prmrec  16254  00lsp  19748  goaln0  32667  bj-pinftynrr  34552  bj-minftynrr  34556  eliuniincex  41607  eliincex  41608  salgencntex  42849  nfermltl2rev  44127
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