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Theorem eqneltri 38272
 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
3 eleq1 2676 . . 3 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
42, 3ax-mp 5 . 2 (𝐴𝐶𝐵𝐶)
51, 4mtbir 312 1 ¬ 𝐴𝐶
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606 This theorem is referenced by:  eliuniincex  38323  eliincex  38324  salgencntex  39237
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