Theorem nelneq 2271

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 2233	. . 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 630	1 |- ( ( A e. C /\ -. B e. C ) -> -. A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166

This theorem is referenced by: (None)