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

Theorem elndif 3431
Description: A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
Assertion
Ref Expression
elndif  |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )

Proof of Theorem elndif
StepHypRef Expression
1 eldifn 3430 . 2  |-  ( A  e.  ( C  \  B )  ->  -.  A  e.  B )
21con2i 114 1  |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1721    \ cdif 3277
This theorem is referenced by:  peano5  4827  undifixp  7057  cantnfreslem  7587  ssfin4  8146  isf32lem3  8191  isf34lem4  8213  xrinfmss  10844  restntr  17200  cmpcld  17419  reconnlem2  18811  lebnumlem1  18939  i1fd  19526  dfon2lem6  25358  onsucconi  26091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-dif 3283
  Copyright terms: Public domain W3C validator