MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqneltrd Structured version   Unicode version

Theorem eqneltrd 2558
Description: If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
eqneltrd.1  |-  ( ph  ->  A  =  B )
eqneltrd.2  |-  ( ph  ->  -.  B  e.  C
)
Assertion
Ref Expression
eqneltrd  |-  ( ph  ->  -.  A  e.  C
)

Proof of Theorem eqneltrd
StepHypRef Expression
1 eqneltrd.2 . 2  |-  ( ph  ->  -.  B  e.  C
)
2 eqneltrd.1 . . 3  |-  ( ph  ->  A  =  B )
32eleq1d 2519 . 2  |-  ( ph  ->  ( A  e.  C  <->  B  e.  C ) )
41, 3mtbird 301 1  |-  ( ph  ->  -.  A  e.  C
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-cleq 2443  df-clel 2446
This theorem is referenced by:  eqneltrrd  2559  omopth2  7120  fpwwe2  8908  sqrneglem  12855  mreexmrid  14680  mplcoe1  17648  mplcoe5  17652  mplcoe2OLD  17654  islln2a  33464  islpln2a  33495  islvol2aN  33539
  Copyright terms: Public domain W3C validator