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Theorem elndif 3300
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 3299 . 2  |-  ( A  e.  ( C  \  B )  ->  -.  A  e.  B )
21con2i 112 1  |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1684    \ cdif 3149
This theorem is referenced by:  peano5  4679  fsnunf2  5719  undifixp  6852  cantnfreslem  7377  dfac9  7762  ssfin4  7936  isf32lem3  7981  isf34lem4  8003  xrinfmss  10628  restntr  16912  cmpcld  17129  reconnlem2  18332  lebnumlem1  18459  i1fd  19036  dfon2lem6  24144  onsucconi  24876  uvcff  27240  islindf4  27308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155
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