Theorem elndif 3252

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 3251	. 2 |- ( A e. ( C \ B ) -> -. A e. B )
2	1	con2i 623	1 |- ( A e. B -> -. A e. ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2142   \ cdif 3119

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-dif 3124

This theorem is referenced by:  ddifnel  3259  inssdif  3364