Theorem elndif 3734

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

Step	Hyp	Ref	Expression
1		eldifn 3733	. 2 |- ( A e. ( C \ B ) -> -. A e. B )
2	1	con2i 134	1 |- ( A e. B -> -. A e. ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   \ cdif 3571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577

This theorem is referenced by:  peano5  7089  extmptsuppeq  7319  undifixp  7944  ssfin4  9132  isf32lem3  9177  isf34lem4  9199  xrinfmss  12140  restntr  20986  cmpcld  21205  reconnlem2  22630  lebnumlem1  22760  i1fd  23448  hgt750lemd  30726  dfon2lem6  31693  onsucconni  32436  meaiininclem  40700  caragendifcl  40728