MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neleqtrd Structured version   Visualization version   GIF version

Theorem neleqtrd 2853
Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
neleqtrd.1 (𝜑 → ¬ 𝐶𝐴)
neleqtrd.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
neleqtrd (𝜑 → ¬ 𝐶𝐵)

Proof of Theorem neleqtrd
StepHypRef Expression
1 neleqtrd.1 . 2 (𝜑 → ¬ 𝐶𝐴)
2 neleqtrd.2 . . 3 (𝜑𝐴 = 𝐵)
32eleq2d 2817 . 2 (𝜑 → (𝐶𝐴𝐶𝐵))
41, 3mtbid 323 1 (𝜑 → ¬ 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-cleq 2722  df-clel 2808
This theorem is referenced by:  neleqtrrd  2854  smoord  8367  r1tskina  10779  ofccat  14920  mreexexlem2d  17593  opptgdim2  28263  acopyeu  28352  dochnel  40567  stoweidlem26  45040  fourierdlem60  45180  fourierdlem61  45181  sge00  45390  sge0sn  45393  sge0split  45423
  Copyright terms: Public domain W3C validator