Theorem elndif 3257

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 3256	. 2 |- ( A e. ( C \ B ) -> -. A e. B )
2	1	con2i 627	1 |- ( A e. B -> -. A e. ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2146   \ cdif 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-dif 3129

This theorem is referenced by:  ddifnel  3264  inssdif  3369