Theorem elndif 3245

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 3244	. 2 |- ( A e. ( C \ B ) -> -. A e. B )
2	1	con2i 617	1 |- ( A e. B -> -. A e. ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2136   \ cdif 3112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-dif 3117

This theorem is referenced by:  ddifnel  3252  inssdif  3357