Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elnelneq2d Structured version   Visualization version   GIF version

Theorem elnelneq2d 40828
Description: Two classes are not equal if one but not the other is an element of a given class. (Contributed by Rohan Ridenour, 12-Aug-2023.)
Hypotheses
Ref Expression
elnelneq2d.1 (𝜑𝐴𝐶)
elnelneq2d.2 (𝜑 → ¬ 𝐵𝐶)
Assertion
Ref Expression
elnelneq2d (𝜑 → ¬ 𝐴 = 𝐵)

Proof of Theorem elnelneq2d
StepHypRef Expression
1 elnelneq2d.2 . 2 (𝜑 → ¬ 𝐵𝐶)
2 simpr 488 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
3 elnelneq2d.1 . . . 4 (𝜑𝐴𝐶)
43adantr 484 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴𝐶)
52, 4eqeltrrd 2917 . 2 ((𝜑𝐴 = 𝐵) → 𝐵𝐶)
61, 5mtand 815 1 (𝜑 → ¬ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = 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:  mnurndlem1  40909
  Copyright terms: Public domain W3C validator