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	syl6ib	|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A ( ( A C_ C /\ C C_ B ) -> ( C = A \/ C = B ) ) ) )