This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cvnbtwn4

Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of MaedaMaeda p. 31. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion cvnbtwn4 
|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A  ( ( A C_ C /\ C C_ B ) -> ( C = A \/ 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 = A \/ C = B ) ) <-> -. ( ( A C_ C /\ C C_ B ) /\ -. ( C = A \/ C = B ) ) )
3 an4 
 |-  ( ( ( A C_ C /\ C C_ B ) /\ ( -. A = C /\ -. C = B ) ) <-> ( ( A C_ C /\ -. A = C ) /\ ( C C_ B /\ -. C = B ) ) )
4 ioran 
 |-  ( -. ( C = A \/ C = B ) <-> ( -. C = A /\ -. C = B ) )
5 eqcom 
 |-  ( C = A <-> A = C )
6 5 notbii 
 |-  ( -. C = A <-> -. A = C )
7 6 anbi1i 
 |-  ( ( -. C = A /\ -. C = B ) <-> ( -. A = C /\ -. C = B ) )
8 4 7 bitri 
 |-  ( -. ( C = A \/ C = B ) <-> ( -. A = C /\ -. C = B ) )
9 8 anbi2i 
 |-  ( ( ( A C_ C /\ C C_ B ) /\ -. ( C = A \/ C = B ) ) <-> ( ( A C_ C /\ C C_ B ) /\ ( -. A = C /\ -. C = B ) ) )
10 dfpss2 
 |-  ( A C. C <-> ( A C_ C /\ -. A = C ) )
11 dfpss2 
 |-  ( C C. B <-> ( C C_ B /\ -. C = B ) )
12 10 11 anbi12i 
 |-  ( ( A C. C /\ C C. B ) <-> ( ( A C_ C /\ -. A = C ) /\ ( C C_ B /\ -. C = B ) ) )
13 3 9 12 3bitr4i 
 |-  ( ( ( A C_ C /\ C C_ B ) /\ -. ( C = A \/ C = B ) ) <-> ( A C. C /\ C C. B ) )
14 13 notbii 
 |-  ( -. ( ( A C_ C /\ C C_ B ) /\ -. ( C = A \/ C = B ) ) <-> -. ( A C. C /\ C C. B ) )
15 2 14 bitr2i 
 |-  ( -. ( A C. C /\ C C. B ) <-> ( ( A C_ C /\ C C_ B ) -> ( C = A \/ C = B ) ) )
16 1 15 imbitrdi 
 |-  ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A  ( ( A C_ C /\ C C_ B ) -> ( C = A \/ C = B ) ) ) )