Theorem nelneq2 2298

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 2260	. . 3 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 159	. 2 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	con3dimp 636	1 |- ( ( A e. B /\ -. A e. C ) -> -. B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  dtruarb  4225  fzneuz  10193