Theorem nelneq2 2259

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 2221	. . 3 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 158	. 2 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	con3dimp 625	1 |- ( ( A e. B /\ -. A e. C ) -> -. B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128

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-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-cleq 2150  df-clel 2153

This theorem is referenced by:  dtruarb  4152  fzneuz  9996