Theorem cvrnbtwn4 16996

Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Th. cvnbtwn4 11861 analog.)

Hypotheses

Ref	Expression
cvrle.b	|- B = (base` K)
cvrle.l	|- L = (le` K)
cvrle.c	|- C = ( <oNEW ` K)

Assertion

Ref	Expression
cvrnbtwn4	|- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> ((XLZ /\ ZLY) <-> (X = Z \/ Z = Y)))

Proof of Theorem cvrnbtwn4

Step	Hyp	Ref	Expression
1		cvrle.b	. . . 4 |- B = (base` K)
2		eqid 1884	. . . 4 |- (lt` K) = (lt` K)
3		cvrle.c	. . . 4 |- C = ( <oNEW ` K)
4	1, 2, 3	cvrnbtwn 16990	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> -. (X(lt` K)Z /\ Z(lt` K)Y))
5		cvrle.l	. . . . . . . . . 10 |- L = (le` K)
6	5, 2	pltval 16781	. . . . . . . . 9 |- ((K e. PosetNEW /\ X e. B /\ Z e. B) -> (X(lt` K)Z <-> (XLZ /\ X =/= Z)))
7	6	3adant3r2 1078	. . . . . . . 8 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (X(lt` K)Z <-> (XLZ /\ X =/= Z)))
8	5, 2	pltval 16781	. . . . . . . . . 10 |- ((K e. PosetNEW /\ Z e. B /\ Y e. B) -> (Z(lt` K)Y <-> (ZLY /\ Z =/= Y)))
9	8	3com23 1074	. . . . . . . . 9 |- ((K e. PosetNEW /\ Y e. B /\ Z e. B) -> (Z(lt` K)Y <-> (ZLY /\ Z =/= Y)))
10	9	3adant3r1 1077	. . . . . . . 8 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (Z(lt` K)Y <-> (ZLY /\ Z =/= Y)))
11	7, 10	anbi12d 690	. . . . . . 7 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((X(lt` K)Z /\ Z(lt` K)Y) <-> ((XLZ /\ X =/= Z) /\ (ZLY /\ Z =/= Y))))
12		neanior 2097	. . . . . . . . 9 |- ((X =/= Z /\ Z =/= Y) <-> -. (X = Z \/ Z = Y))
13	12	anbi2i 538	. . . . . . . 8 |- (((XLZ /\ ZLY) /\ (X =/= Z /\ Z =/= Y)) <-> ((XLZ /\ ZLY) /\ -. (X = Z \/ Z = Y)))
14		an4 564	. . . . . . . 8 |- (((XLZ /\ ZLY) /\ (X =/= Z /\ Z =/= Y)) <-> ((XLZ /\ X =/= Z) /\ (ZLY /\ Z =/= Y)))
15	13, 14	bitr3i 192	. . . . . . 7 |- (((XLZ /\ ZLY) /\ -. (X = Z \/ Z = Y)) <-> ((XLZ /\ X =/= Z) /\ (ZLY /\ Z =/= Y)))
16	11, 15	syl6rbbr 598	. . . . . 6 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (((XLZ /\ ZLY) /\ -. (X = Z \/ Z = Y)) <-> (X(lt` K)Z /\ Z(lt` K)Y)))
17	16	notbid 673	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (-. ((XLZ /\ ZLY) /\ -. (X = Z \/ Z = Y)) <-> -. (X(lt` K)Z /\ Z(lt` K)Y)))
18		iman 256	. . . . 5 |- (((XLZ /\ ZLY) -> (X = Z \/ Z = Y)) <-> -. ((XLZ /\ ZLY) /\ -. (X = Z \/ Z = Y)))
19	17, 18	syl5rbb 592	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (-. (X(lt` K)Z /\ Z(lt` K)Y) <-> ((XLZ /\ ZLY) -> (X = Z \/ Z = Y))))
20	19	3adant3 896	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> (-. (X(lt` K)Z /\ Z(lt` K)Y) <-> ((XLZ /\ ZLY) -> (X = Z \/ Z = Y))))
21	4, 20	mpbid 212	. 2 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> ((XLZ /\ ZLY) -> (X = Z \/ Z = Y)))
22		breq1 3341	. . . . 5 |- (X = Z -> (XLZ <-> ZLZ))
23	1, 5	posref 16775	. . . . . . 7 |- ((K e. PosetNEW /\ Z e. B) -> ZLZ)
24	23	3ad2antr3 1043	. . . . . 6 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ZLZ)
25	24	3adant3 896	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> ZLZ)
26	22, 25	syl5cbir 228	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> (X = Z -> XLZ))
27		breq2 3342	. . . . 5 |- (Z = Y -> (XLZ <-> XLY))
28	1, 5, 3	cvrle 16995	. . . . . . . 8 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ XCY) -> XLY)
29	28	ex 402	. . . . . . 7 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (XCY -> XLY))
30	29	3adant3r3 1079	. . . . . 6 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (XCY -> XLY))
31	30	3impia 1064	. . . . 5 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> XLY)
32	27, 31	syl5cbir 228	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> (Z = Y -> XLZ))
33	26, 32	jaod 469	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> ((X = Z \/ Z = Y) -> XLZ))
34		breq1 3341	. . . . 5 |- (X = Z -> (XLY <-> ZLY))
35	34, 31	syl5cbi 226	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> (X = Z -> ZLY))
36		breq2 3342	. . . . 5 |- (Z = Y -> (ZLZ <-> ZLY))
37	36, 25	syl5cbi 226	. . . 4 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> (Z = Y -> ZLY))
38	35, 37	jaod 469	. . 3 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> ((X = Z \/ Z = Y) -> ZLY))
39	33, 38	jcad 661	. 2 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> ((X = Z \/ Z = Y) -> (XLZ /\ ZLY)))
40	21, 39	impbid 574	1 |- ((K e. PosetNEW /\ (X e. B /\ Y e. B /\ Z e. B) /\ XCY) -> ((XLZ /\ ZLY) <-> (X = Z \/ Z = Y)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  basecbs 16758  lecple 16759  PosetNEWcpo 16760  ltcplt 16761   <o_NEW ccvr 16980

This theorem is referenced by:  cvrcmp 16999  leatom 17005

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  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-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-mpt 5006  df-struct 16708  df-poset 16772  df-plt 16780  df-covers 16984

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