MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  neleq2 Structured version   Unicode version

Theorem neleq2 2702
Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
Assertion
Ref Expression
neleq2  |-  ( A  =  B  ->  ( C  e/  A  <->  C  e/  B ) )

Proof of Theorem neleq2
StepHypRef Expression
1 eleq2 2499 . . 3  |-  ( A  =  B  ->  ( C  e.  A  <->  C  e.  B ) )
21notbid 287 . 2  |-  ( A  =  B  ->  ( -.  C  e.  A  <->  -.  C  e.  B ) )
3 df-nel 2604 . 2  |-  ( C  e/  A  <->  -.  C  e.  A )
4 df-nel 2604 . 2  |-  ( C  e/  B  <->  -.  C  e.  B )
52, 3, 43bitr4g 281 1  |-  ( A  =  B  ->  ( C  e/  A  <->  C  e/  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    e/ wnel 2602
This theorem is referenced by:  neleq12d  2703  noinfep  7617  isfbas  17866  nbgra0nb  21446  cusgrares  21486  frgrawopreglem4  28510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2431  df-clel 2434  df-nel 2604
  Copyright terms: Public domain W3C validator