Theorem nelneq 2276

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 2238	. . 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 635	1 |- ( ( A e. C /\ -. B e. C ) -> -. A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146

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 614  ax-in2 615  ax-5 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-17 1524  ax-ial 1532  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-cleq 2168  df-clel 2171

This theorem is referenced by: (None)