ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neleqtrrd Unicode version

Theorem neleqtrrd 2152
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
neleqtrrd.1  |-  ( ph  ->  -.  C  e.  B
)
neleqtrrd.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
neleqtrrd  |-  ( ph  ->  -.  C  e.  A
)

Proof of Theorem neleqtrrd
StepHypRef Expression
1 neleqtrrd.1 . 2  |-  ( ph  ->  -.  C  e.  B
)
2 neleqtrrd.2 . . 3  |-  ( ph  ->  A  =  B )
32eleq2d 2123 . 2  |-  ( ph  ->  ( C  e.  A  <->  C  e.  B ) )
41, 3mtbird 608 1  |-  ( ph  ->  -.  C  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1259    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator