Theorem lcvnbtwn2 34314

Description: The covers relation implies no in-betweenness. (cvnbtwn2 29146 analog.) (Contributed by NM, 7-Jan-2015.)

Hypotheses

Ref	Expression
lcvnbtwn.s	|- S = ( LSubSp ` W )
lcvnbtwn.c	|- C = ( <oLL ` W )
lcvnbtwn.w	|- ( ph -> W e. X )
lcvnbtwn.r	|- ( ph -> R e. S )
lcvnbtwn.t	|- ( ph -> T e. S )
lcvnbtwn.u	|- ( ph -> U e. S )
lcvnbtwn.d	|- ( ph -> R C T )
lcvnbtwn2.p	|- ( ph -> R C. U )
lcvnbtwn2.q	|- ( ph -> U C_ T )

Assertion

Ref	Expression
lcvnbtwn2	|- ( ph -> U = T )

Proof of Theorem lcvnbtwn2

Step	Hyp	Ref	Expression
1		lcvnbtwn2.p	. 2 |- ( ph -> R C. U )
2		lcvnbtwn2.q	. 2 |- ( ph -> U C_ T )
3		lcvnbtwn.s	. . . 4 |- S = ( LSubSp ` W )
4		lcvnbtwn.c	. . . 4 |- C = ( <oLL ` W )
5		lcvnbtwn.w	. . . 4 |- ( ph -> W e. X )
6		lcvnbtwn.r	. . . 4 |- ( ph -> R e. S )
7		lcvnbtwn.t	. . . 4 |- ( ph -> T e. S )
8		lcvnbtwn.u	. . . 4 |- ( ph -> U e. S )
9		lcvnbtwn.d	. . . 4 |- ( ph -> R C T )
10	3, 4, 5, 6, 7, 8, 9	lcvnbtwn 34312	. . 3 |- ( ph -> -. ( R C. U /\ U C. T ) )
11		iman 440	. . . 4 |- ( ( ( R C. U /\ U C_ T ) -> U = T ) <-> -. ( ( R C. U /\ U C_ T ) /\ -. U = T ) )
12		anass 681	. . . . . 6 |- ( ( ( R C. U /\ U C_ T ) /\ -. U = T ) <-> ( R C. U /\ ( U C_ T /\ -. U = T ) ) )
13		dfpss2 3692	. . . . . . 7 |- ( U C. T <-> ( U C_ T /\ -. U = T ) )
14	13	anbi2i 730	. . . . . 6 |- ( ( R C. U /\ U C. T ) <-> ( R C. U /\ ( U C_ T /\ -. U = T ) ) )
15	12, 14	bitr4i 267	. . . . 5 |- ( ( ( R C. U /\ U C_ T ) /\ -. U = T ) <-> ( R C. U /\ U C. T ) )
16	15	notbii 310	. . . 4 |- ( -. ( ( R C. U /\ U C_ T ) /\ -. U = T ) <-> -. ( R C. U /\ U C. T ) )
17	11, 16	bitr2i 265	. . 3 |- ( -. ( R C. U /\ U C. T ) <-> ( ( R C. U /\ U C_ T ) -> U = T ) )
18	10, 17	sylib 208	. 2 |- ( ph -> ( ( R C. U /\ U C_ T ) -> U = T ) )
19	1, 2, 18	mp2and 715	1 |- ( ph -> U = T )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   C. wpss 3575   class class class wbr 4653   ` cfv 5888   LSubSpclss 18932   <oL_L clcv 34305

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-pss 3590  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-lcv 34306

This theorem is referenced by:  lcvat  34317  lsatexch  34330