Theorem nelneq 2725

Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)

Assertion

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

Proof of Theorem nelneq

Step	Hyp	Ref	Expression
1		eleq1 2689	. . 3 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	biimpcd 239	. 2 |- ( A e. C -> ( A = B -> B e. C ) )
3	2	con3dimp 457	1 |- ( ( A e. C /\ -. B e. C ) -> -. A = B )

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:  onfununi  7438  suc11reg  8516  cantnfp1lem3  8577  oemapvali  8581  xrge0neqmnf  12276  mreexmrid  16303  supxrnemnf  29534  onint1  32448  maxidln0  33844  rencldnfilem  37384  climlimsupcex  40001  icccncfext  40100