Theorem nelneq2 2726

Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Assertion

Ref	Expression
nelneq2	|- ( ( A e. B /\ -. A e. C ) -> -. B = C )

Proof of Theorem nelneq2

Step	Hyp	Ref	Expression
1		eleq2 2690	. . 3 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 239	. 2 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	con3dimp 457	1 |- ( ( A e. B /\ -. A e. C ) -> -. B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990

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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by:  ssnelpss  3718  opthwiener  4976  ssfin4  9132  pwxpndom2  9487  fzneuz  12421  hauspwpwf1  21791  topdifinffinlem  33195  clsk1indlem1  38343