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Metamath Proof Explorer

Theorem cvnbtwn2

Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion cvnbtwn2 
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A  ( ( A C. C /\ C C_ B ) -> C = B ) ) )

Proof

Step Hyp Ref Expression
1 cvnbtwn 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A  -. ( A C. C /\ C C. B ) ) )
2 iman 
 |-  ( ( ( A C. C /\ C C_ B ) -> C = B ) <-> -. ( ( A C. C /\ C C_ B ) /\ -. C = B ) )
3 anass 
 |-  ( ( ( A C. C /\ C C_ B ) /\ -. C = B ) <-> ( A C. C /\ ( C C_ B /\ -. C = B ) ) )
4 dfpss2 
 |-  ( C C. B <-> ( C C_ B /\ -. C = B ) )
5 4 anbi2i 
 |-  ( ( A C. C /\ C C. B ) <-> ( A C. C /\ ( C C_ B /\ -. C = B ) ) )
6 3 5 bitr4i 
 |-  ( ( ( A C. C /\ C C_ B ) /\ -. C = B ) <-> ( A C. C /\ C C. B ) )
7 6 notbii 
 |-  ( -. ( ( A C. C /\ C C_ B ) /\ -. C = B ) <-> -. ( A C. C /\ C C. B ) )
8 2 7 bitr2i 
 |-  ( -. ( A C. C /\ C C. B ) <-> ( ( A C. C /\ C C_ B ) -> C = B ) )
9 1 8 imbitrdi 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A  ( ( A C. C /\ C C_ B ) -> C = B ) ) )