Theorem nelelne 2892

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 2891	. 2 |- ( ( C e. B /\ -. A e. B ) -> C =/= A )
2	1	expcom 451	1 |- ( -. A e. B -> ( C e. B -> C =/= A ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ne 2795

This theorem is referenced by:  ssdifsn  4318  difsn  4328  frgrncvvdeqlem7  27169  frgrncvvdeqlem9  27171  prter2  34166