Theorem neneor 2893

Description: If two classes are different, a third class must be different of at least one of them. (Contributed by Thierry Arnoux, 8-Aug-2020.)

Assertion

Ref	Expression
neneor	|- ( A =/= B -> ( A =/= C \/ B =/= C ) )

Proof of Theorem neneor

Step	Hyp	Ref	Expression
1		eqtr3 2643	. . 3 |- ( ( A = C /\ B = C ) -> A = B )
2	1	necon3ai 2819	. 2 |- ( A =/= B -> -. ( A = C /\ B = C ) )
3		neorian 2888	. 2 |- ( ( A =/= C \/ B =/= C ) <-> -. ( A = C /\ B = C ) )
4	2, 3	sylibr 224	1 |- ( A =/= B -> ( A =/= C \/ B =/= C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   =/= 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-or 385  df-an 386  df-ex 1705  df-cleq 2615  df-ne 2795

This theorem is referenced by:  trgcopyeulem  25697