Theorem cvrnbtwn4 34566

Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 29148 analog.) (Contributed by NM, 18-Oct-2011.)

Hypotheses

Ref	Expression
cvrle.b	|- B = ( Base ` K )
cvrle.l	|- .<_ = ( le ` K )
cvrle.c	|- C = ( <o ` K )

Assertion

Ref	Expression
cvrnbtwn4	|- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X .<_ Z /\ Z .<_ Y ) <-> ( X = Z \/ Z = Y ) ) )

Proof of Theorem cvrnbtwn4

Step	Hyp	Ref	Expression
1		cvrle.b	. . . 4 |- B = ( Base ` K )
2		eqid 2622	. . . 4 |- ( lt ` K ) = ( lt ` K )
3		cvrle.c	. . . 4 |- C = ( <o ` K )
4	1, 2, 3	cvrnbtwn 34558	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> -. ( X ( lt ` K ) Z /\ Z ( lt ` K ) Y ) )
5		iman 440	. . . . 5 |- ( ( ( X .<_ Z /\ Z .<_ Y ) -> ( X = Z \/ Z = Y ) ) <-> -. ( ( X .<_ Z /\ Z .<_ Y ) /\ -. ( X = Z \/ Z = Y ) ) )
6		cvrle.l	. . . . . . . . . 10 |- .<_ = ( le ` K )
7	6, 2	pltval 16960	. . . . . . . . 9 |- ( ( K e. Poset /\ X e. B /\ Z e. B ) -> ( X ( lt ` K ) Z <-> ( X .<_ Z /\ X =/= Z ) ) )
8	7	3adant3r2 1275	. . . . . . . 8 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X ( lt ` K ) Z <-> ( X .<_ Z /\ X =/= Z ) ) )
9	6, 2	pltval 16960	. . . . . . . . . 10 |- ( ( K e. Poset /\ Z e. B /\ Y e. B ) -> ( Z ( lt ` K ) Y <-> ( Z .<_ Y /\ Z =/= Y ) ) )
10	9	3com23 1271	. . . . . . . . 9 |- ( ( K e. Poset /\ Y e. B /\ Z e. B ) -> ( Z ( lt ` K ) Y <-> ( Z .<_ Y /\ Z =/= Y ) ) )
11	10	3adant3r1 1274	. . . . . . . 8 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Z ( lt ` K ) Y <-> ( Z .<_ Y /\ Z =/= Y ) ) )
12	8, 11	anbi12d 747	. . . . . . 7 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ( lt ` K ) Z /\ Z ( lt ` K ) Y ) <-> ( ( X .<_ Z /\ X =/= Z ) /\ ( Z .<_ Y /\ Z =/= Y ) ) ) )
13		neanior 2886	. . . . . . . . 9 |- ( ( X =/= Z /\ Z =/= Y ) <-> -. ( X = Z \/ Z = Y ) )
14	13	anbi2i 730	. . . . . . . 8 |- ( ( ( X .<_ Z /\ Z .<_ Y ) /\ ( X =/= Z /\ Z =/= Y ) ) <-> ( ( X .<_ Z /\ Z .<_ Y ) /\ -. ( X = Z \/ Z = Y ) ) )
15		an4 865	. . . . . . . 8 |- ( ( ( X .<_ Z /\ Z .<_ Y ) /\ ( X =/= Z /\ Z =/= Y ) ) <-> ( ( X .<_ Z /\ X =/= Z ) /\ ( Z .<_ Y /\ Z =/= Y ) ) )
16	14, 15	bitr3i 266	. . . . . . 7 |- ( ( ( X .<_ Z /\ Z .<_ Y ) /\ -. ( X = Z \/ Z = Y ) ) <-> ( ( X .<_ Z /\ X =/= Z ) /\ ( Z .<_ Y /\ Z =/= Y ) ) )
17	12, 16	syl6rbbr 279	. . . . . 6 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( X .<_ Z /\ Z .<_ Y ) /\ -. ( X = Z \/ Z = Y ) ) <-> ( X ( lt ` K ) Z /\ Z ( lt ` K ) Y ) ) )
18	17	notbid 308	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( -. ( ( X .<_ Z /\ Z .<_ Y ) /\ -. ( X = Z \/ Z = Y ) ) <-> -. ( X ( lt ` K ) Z /\ Z ( lt ` K ) Y ) ) )
19	5, 18	syl5rbb 273	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( -. ( X ( lt ` K ) Z /\ Z ( lt ` K ) Y ) <-> ( ( X .<_ Z /\ Z .<_ Y ) -> ( X = Z \/ Z = Y ) ) ) )
20	19	3adant3 1081	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( -. ( X ( lt ` K ) Z /\ Z ( lt ` K ) Y ) <-> ( ( X .<_ Z /\ Z .<_ Y ) -> ( X = Z \/ Z = Y ) ) ) )
21	4, 20	mpbid 222	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X .<_ Z /\ Z .<_ Y ) -> ( X = Z \/ Z = Y ) ) )
22	1, 6	posref 16951	. . . . . . 7 |- ( ( K e. Poset /\ Z e. B ) -> Z .<_ Z )
23	22	3ad2antr3 1228	. . . . . 6 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z .<_ Z )
24	23	3adant3 1081	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> Z .<_ Z )
25		breq1 4656	. . . . 5 |- ( X = Z -> ( X .<_ Z <-> Z .<_ Z ) )
26	24, 25	syl5ibrcom 237	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( X = Z -> X .<_ Z ) )
27	1, 6, 3	cvrle 34565	. . . . . . . 8 |- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .<_ Y )
28	27	ex 450	. . . . . . 7 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X C Y -> X .<_ Y ) )
29	28	3adant3r3 1276	. . . . . 6 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Y -> X .<_ Y ) )
30	29	3impia 1261	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> X .<_ Y )
31		breq2 4657	. . . . 5 |- ( Z = Y -> ( X .<_ Z <-> X .<_ Y ) )
32	30, 31	syl5ibrcom 237	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( Z = Y -> X .<_ Z ) )
33	26, 32	jaod 395	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X = Z \/ Z = Y ) -> X .<_ Z ) )
34		breq1 4656	. . . . 5 |- ( X = Z -> ( X .<_ Y <-> Z .<_ Y ) )
35	30, 34	syl5ibcom 235	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( X = Z -> Z .<_ Y ) )
36		breq2 4657	. . . . 5 |- ( Z = Y -> ( Z .<_ Z <-> Z .<_ Y ) )
37	24, 36	syl5ibcom 235	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( Z = Y -> Z .<_ Y ) )
38	35, 37	jaod 395	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X = Z \/ Z = Y ) -> Z .<_ Y ) )
39	33, 38	jcad 555	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X = Z \/ Z = Y ) -> ( X .<_ Z /\ Z .<_ Y ) ) )
40	21, 39	impbid 202	1 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X .<_ Z /\ Z .<_ Y ) <-> ( X = Z \/ Z = Y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   ltcplt 16941   <o ccvr 34549

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-preset 16928  df-poset 16946  df-plt 16958  df-covers 34553

This theorem is referenced by:  cvrcmp  34570  leatb  34579  2llnmat  34810  2lnat  35070