MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nelneq2 Structured version   Unicode version

Theorem nelneq2 2526
Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
nelneq2  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )

Proof of Theorem nelneq2
StepHypRef Expression
1 eleq2 2490 . . 3  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 227 . 2  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32con3dimp 442 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-cleq 2416  df-clel 2419
This theorem is referenced by:  ssnelpss  3797  opthwiener  4660  ssfin4  8686  pwxpndom2  9036  fzneuz  11821  hauspwpwf1  20939  vdgr1b  25569  topdifinffinlem  31657
  Copyright terms: Public domain W3C validator