Theorem nelelne 2432

Description: Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)

Assertion

Ref	Expression
nelelne	|- ( -. A e. B -> ( C e. B -> C =/= A ) )

Proof of Theorem nelelne

Step	Hyp	Ref	Expression
1		nelne2 2431	. 2 |- ( ( C e. B /\ -. A e. B ) -> C =/= A )
2	1	expcom 115	1 |- ( -. A e. B -> ( C e. B -> C =/= A ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2141   =/= wne 2340

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  df-ne 2341

This theorem is referenced by: (None)