Theorem nelneq 2294

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 2256	. . 3 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	biimpcd 159	. 2 |- ( A e. C -> ( A = B -> B e. C ) )
3	2	con3dimp 636	1 |- ( ( A e. C /\ -. B e. C ) -> -. A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by: (None)