Theorem cvrnbtwn 34558

Description: There is no element between the two arguments of the covers relation. (cvnbtwn 29145 analog.) (Contributed by NM, 18-Oct-2011.)

Hypotheses

Ref	Expression
cvrfval.b	|- B = ( Base ` K )
cvrfval.s	|- .< = ( lt ` K )
cvrfval.c	|- C = ( <o ` K )

Assertion

Ref	Expression
cvrnbtwn	|- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> -. ( X .< Z /\ Z .< Y ) )

Proof of Theorem cvrnbtwn

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvrfval.b	. . . . 5 |- B = ( Base ` K )
2		cvrfval.s	. . . . 5 |- .< = ( lt ` K )
3		cvrfval.c	. . . . 5 |- C = ( <o ` K )
4	1, 2, 3	cvrval 34556	. . . 4 |- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
5	4	3adant3r3 1276	. . 3 |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Y <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
6		ralnex 2992	. . . . . . 7 |- ( A. z e. B -. ( X .< z /\ z .< Y ) <-> -. E. z e. B ( X .< z /\ z .< Y ) )
7		breq2 4657	. . . . . . . . . 10 |- ( z = Z -> ( X .< z <-> X .< Z ) )
8		breq1 4656	. . . . . . . . . 10 |- ( z = Z -> ( z .< Y <-> Z .< Y ) )
9	7, 8	anbi12d 747	. . . . . . . . 9 |- ( z = Z -> ( ( X .< z /\ z .< Y ) <-> ( X .< Z /\ Z .< Y ) ) )
10	9	notbid 308	. . . . . . . 8 |- ( z = Z -> ( -. ( X .< z /\ z .< Y ) <-> -. ( X .< Z /\ Z .< Y ) ) )
11	10	rspcv 3305	. . . . . . 7 |- ( Z e. B -> ( A. z e. B -. ( X .< z /\ z .< Y ) -> -. ( X .< Z /\ Z .< Y ) ) )
12	6, 11	syl5bir 233	. . . . . 6 |- ( Z e. B -> ( -. E. z e. B ( X .< z /\ z .< Y ) -> -. ( X .< Z /\ Z .< Y ) ) )
13	12	adantld 483	. . . . 5 |- ( Z e. B -> ( ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) -> -. ( X .< Z /\ Z .< Y ) ) )
14	13	3ad2ant3 1084	. . . 4 |- ( ( X e. B /\ Y e. B /\ Z e. B ) -> ( ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) -> -. ( X .< Z /\ Z .< Y ) ) )
15	14	adantl 482	. . 3 |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) -> -. ( X .< Z /\ Z .< Y ) ) )
16	5, 15	sylbid 230	. 2 |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Y -> -. ( X .< Z /\ Z .< Y ) ) )
17	16	3impia 1261	1 |- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ X C Y ) -> -. ( X .< Z /\ Z .< Y ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888   Basecbs 15857   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-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-covers 34553

This theorem is referenced by:  cvrnbtwn2  34562  cvrnbtwn3  34563  cvrnbtwn4  34566  ltltncvr  34709