Theorem nelneq 2189

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 2151	. . 3 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	biimpcd 158	. 2 |- ( A e. C -> ( A = B -> B e. C ) )
3	2	con3dimp 600	1 |- ( ( A e. C /\ -. B e. C ) -> -. A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082  df-clel 2085

This theorem is referenced by: (None)