Theorem cvnbtwn 27939

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

Assertion

Ref	Expression
cvnbtwn	|- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH B -> -. ( A C. C /\ C C. B ) ) )

Proof of Theorem cvnbtwn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvbr 27935	. . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B <-> ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) ) )
2		psseq2 3521	. . . . . . . . 9 |- ( x = C -> ( A C. x <-> A C. C ) )
3		psseq1 3520	. . . . . . . . 9 |- ( x = C -> ( x C. B <-> C C. B ) )
4	2, 3	anbi12d 717	. . . . . . . 8 |- ( x = C -> ( ( A C. x /\ x C. B ) <-> ( A C. C /\ C C. B ) ) )
5	4	rspcev 3150	. . . . . . 7 |- ( ( C e. CH /\ ( A C. C /\ C C. B ) ) -> E. x e. CH ( A C. x /\ x C. B ) )
6	5	ex 436	. . . . . 6 |- ( C e. CH -> ( ( A C. C /\ C C. B ) -> E. x e. CH ( A C. x /\ x C. B ) ) )
7	6	con3rr3 142	. . . . 5 |- ( -. E. x e. CH ( A C. x /\ x C. B ) -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) )
8	7	adantl 468	. . . 4 |- ( ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) )
9	1, 8	syl6bi 232	. . 3 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) ) )
10	9	com23 81	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( C e. CH -> ( A <oH B -> -. ( A C. C /\ C C. B ) ) ) )
11	10	3impia 1205	1 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH B -> -. ( A C. C /\ C C. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   E.wrex 2738   C. wpss 3405   class class class wbr 4402   CHcch 26582   <oH ccv 26617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-cv 27932

This theorem is referenced by:  cvnbtwn2  27940  cvnbtwn3  27941  cvnbtwn4  27942  cvntr  27945