Theorem nelelne 2468

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 2467	. 2 |- ( ( C e. B /\ -. A e. B ) -> C =/= A )
2	1	expcom 116	1 |- ( -. A e. B -> ( C e. B -> C =/= A ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2176   =/= wne 2376

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 615  ax-in2 616  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201  df-ne 2377

This theorem is referenced by: (None)