Theorem cvnbtwn 11858

Description: The covers relation implies no in-betweenness.

Assertion

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

Proof of Theorem cvnbtwn

Step	Hyp	Ref	Expression
1		cvbr 11854	. . . 4 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A C. B /\ -. E.x e. CH (A C. x /\ x C. B))))
2		psseq2 2698	. . . . . . . . . 10 |- (x = C -> (A C. x <-> A C. C))
3		psseq1 2697	. . . . . . . . . 10 |- (x = C -> (x C. B <-> C C. B))
4	2, 3	anbi12d 690	. . . . . . . . 9 |- (x = C -> ((A C. x /\ x C. B) <-> (A C. C /\ C C. B)))
5	4	rcla4ev 2381	. . . . . . . 8 |- ((C e. CH /\ (A C. C /\ C C. B)) -> E.x e. CH (A C. x /\ x C. B))
6	5	ex 402	. . . . . . 7 |- (C e. CH -> ((A C. C /\ C C. B) -> E.x e. CH (A C. x /\ x C. B)))
7	6	con3d 111	. . . . . 6 |- (C e. CH -> (-. E.x e. CH (A C. x /\ x C. B) -> -. (A C. C /\ C C. B)))
8	7	com12 14	. . . . 5 |- (-. E.x e. CH (A C. x /\ x C. B) -> (C e. CH -> -. (A C. C /\ C C. B)))
9	8	adantl 424	. . . 4 |- ((A C. B /\ -. E.x e. CH (A C. x /\ x C. B)) -> (C e. CH -> -. (A C. C /\ C C. B)))
10	1, 9	syl6bi 231	. . 3 |- ((A e. CH /\ B e. CH) -> (A <o B -> (C e. CH -> -. (A C. C /\ C C. B))))
11	10	com23 36	. 2 |- ((A e. CH /\ B e. CH) -> (C e. CH -> (A <o B -> -. (A C. C /\ C C. B))))
12	11	3impia 1064	1 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> -. (A C. C /\ C C. B)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106   C. wpss 2594   class class class wbr 3338  CHcch 10430   <o ccv 10466

This theorem is referenced by:  cvnbtwn2 11859  cvnbtwn3 11860  cvnbtwn4 11861  cvntr 11864

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-cv 11851

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