Theorem cvnbtwn 29145

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 29141	. . . 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 3695	. . . . . . . . 9 |- ( x = C -> ( A C. x <-> A C. C ) )
3		psseq1 3694	. . . . . . . . 9 |- ( x = C -> ( x C. B <-> C C. B ) )
4	2, 3	anbi12d 747	. . . . . . . 8 |- ( x = C -> ( ( A C. x /\ x C. B ) <-> ( A C. C /\ C C. B ) ) )
5	4	rspcev 3309	. . . . . . 7 |- ( ( C e. CH /\ ( A C. C /\ C C. B ) ) -> E. x e. CH ( A C. x /\ x C. B ) )
6	5	ex 450	. . . . . 6 |- ( C e. CH -> ( ( A C. C /\ C C. B ) -> E. x e. CH ( A C. x /\ x C. B ) ) )
7	6	con3rr3 151	. . . . 5 |- ( -. E. x e. CH ( A C. x /\ x C. B ) -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) )
8	7	adantl 482	. . . 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 243	. . 3 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) ) )
10	9	com23 86	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( C e. CH -> ( A <oH B -> -. ( A C. C /\ C C. B ) ) ) )
11	10	3impia 1261	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C. wpss 3575   class class class wbr 4653   CHcch 27786   <oH ccv 27821

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cv 29138

This theorem is referenced by:  cvnbtwn2  29146  cvnbtwn3  29147  cvnbtwn4  29148  cvntr  29151