Theorem cvrnbtwn3 34563

Description: The covers relation implies no in-betweenness. (cvnbtwn3 29147 analog.) (Contributed by NM, 4-Nov-2011.)

Hypotheses

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

Assertion

Ref	Expression
cvrnbtwn3	|- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X .<_ Z /\ Z .< Y ) <-> X = Z ) )

Proof of Theorem cvrnbtwn3

Step	Hyp	Ref	Expression
1		cvrletr.b	. . . 4 |- B = ( Base ` K )
2		cvrletr.s	. . . 4 |- .< = ( lt ` K )
3		cvrletr.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 .< Z /\ Z .< Y ) )
5		cvrletr.l	. . . . . . . . 9 |- .<_ = ( le ` K )
6	5, 2	pltval 16960	. . . . . . . 8 |- ( ( K e. Poset /\ X e. B /\ Z e. B ) -> ( X .< Z <-> ( X .<_ Z /\ X =/= Z ) ) )
7	6	3adant3r2 1275	. . . . . . 7 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .< Z <-> ( X .<_ Z /\ X =/= Z ) ) )
8	7	3adant3 1081	. . . . . 6 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( X .< Z <-> ( X .<_ Z /\ X =/= Z ) ) )
9	8	anbi1d 741	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X .< Z /\ Z .< Y ) <-> ( ( X .<_ Z /\ X =/= Z ) /\ Z .< Y ) ) )
10	9	notbid 308	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( -. ( X .< Z /\ Z .< Y ) <-> -. ( ( X .<_ Z /\ X =/= Z ) /\ Z .< Y ) ) )
11		an32 839	. . . . . . 7 |- ( ( ( X .<_ Z /\ X =/= Z ) /\ Z .< Y ) <-> ( ( X .<_ Z /\ Z .< Y ) /\ X =/= Z ) )
12		df-ne 2795	. . . . . . . 8 |- ( X =/= Z <-> -. X = Z )
13	12	anbi2i 730	. . . . . . 7 |- ( ( ( X .<_ Z /\ Z .< Y ) /\ X =/= Z ) <-> ( ( X .<_ Z /\ Z .< Y ) /\ -. X = Z ) )
14	11, 13	bitri 264	. . . . . 6 |- ( ( ( X .<_ Z /\ X =/= Z ) /\ Z .< Y ) <-> ( ( X .<_ Z /\ Z .< Y ) /\ -. X = Z ) )
15	14	notbii 310	. . . . 5 |- ( -. ( ( X .<_ Z /\ X =/= Z ) /\ Z .< Y ) <-> -. ( ( X .<_ Z /\ Z .< Y ) /\ -. X = Z ) )
16		iman 440	. . . . 5 |- ( ( ( X .<_ Z /\ Z .< Y ) -> X = Z ) <-> -. ( ( X .<_ Z /\ Z .< Y ) /\ -. X = Z ) )
17	15, 16	bitr4i 267	. . . 4 |- ( -. ( ( X .<_ Z /\ X =/= Z ) /\ Z .< Y ) <-> ( ( X .<_ Z /\ Z .< Y ) -> X = Z ) )
18	10, 17	syl6bb 276	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( -. ( X .< Z /\ Z .< Y ) <-> ( ( X .<_ Z /\ Z .< Y ) -> X = Z ) ) )
19	4, 18	mpbid 222	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X .<_ Z /\ Z .< Y ) -> X = Z ) )
20	1, 5	posref 16951	. . . . . 6 |- ( ( K e. Poset /\ X e. B ) -> X .<_ X )
21		breq2 4657	. . . . . 6 |- ( X = Z -> ( X .<_ X <-> X .<_ Z ) )
22	20, 21	syl5ibcom 235	. . . . 5 |- ( ( K e. Poset /\ X e. B ) -> ( X = Z -> X .<_ Z ) )
23	22	3ad2antr1 1226	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X = Z -> X .<_ Z ) )
24	23	3adant3 1081	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( X = Z -> X .<_ Z ) )
25		simp1 1061	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> K e. Poset )
26		simp21 1094	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> X e. B )
27		simp22 1095	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> Y e. B )
28		simp3 1063	. . . . 5 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> X C Y )
29	1, 2, 3	cvrlt 34557	. . . . 5 |- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .< Y )
30	25, 26, 27, 28, 29	syl31anc 1329	. . . 4 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> X .< Y )
31		breq1 4656	. . . 4 |- ( X = Z -> ( X .< Y <-> Z .< Y ) )
32	30, 31	syl5ibcom 235	. . 3 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( X = Z -> Z .< Y ) )
33	24, 32	jcad 555	. 2 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( X = Z -> ( X .<_ Z /\ Z .< Y ) ) )
34	19, 33	impbid 202	1 |- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> ( ( X .<_ Z /\ Z .< Y ) <-> X = Z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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:  atcvreq0  34601  cvratlem  34707