Theorem lcvnbtwn 34312

Description: The covers relation implies no in-betweenness. (cvnbtwn 29145 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 )

Assertion

Ref	Expression
lcvnbtwn	|- ( ph -> -. ( R C. U /\ U C. T ) )

Proof of Theorem lcvnbtwn

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcvnbtwn.d	. . . 4 |- ( ph -> R C T )
2		lcvnbtwn.s	. . . . 5 |- S = ( LSubSp ` W )
3		lcvnbtwn.c	. . . . 5 |- C = ( <oLL ` W )
4		lcvnbtwn.w	. . . . 5 |- ( ph -> W e. X )
5		lcvnbtwn.r	. . . . 5 |- ( ph -> R e. S )
6		lcvnbtwn.t	. . . . 5 |- ( ph -> T e. S )
7	2, 3, 4, 5, 6	lcvbr 34308	. . . 4 |- ( ph -> ( R C T <-> ( R C. T /\ -. E. u e. S ( R C. u /\ u C. T ) ) ) )
8	1, 7	mpbid 222	. . 3 |- ( ph -> ( R C. T /\ -. E. u e. S ( R C. u /\ u C. T ) ) )
9	8	simprd 479	. 2 |- ( ph -> -. E. u e. S ( R C. u /\ u C. T ) )
10		lcvnbtwn.u	. . 3 |- ( ph -> U e. S )
11		psseq2 3695	. . . . 5 |- ( u = U -> ( R C. u <-> R C. U ) )
12		psseq1 3694	. . . . 5 |- ( u = U -> ( u C. T <-> U C. T ) )
13	11, 12	anbi12d 747	. . . 4 |- ( u = U -> ( ( R C. u /\ u C. T ) <-> ( R C. U /\ U C. T ) ) )
14	13	rspcev 3309	. . 3 |- ( ( U e. S /\ ( R C. U /\ U C. T ) ) -> E. u e. S ( R C. u /\ u C. T ) )
15	10, 14	sylan 488	. 2 |- ( ( ph /\ ( R C. U /\ U C. T ) ) -> E. u e. S ( R C. u /\ u C. T ) )
16	9, 15	mtand 691	1 |- ( ph -> -. ( R C. U /\ U C. T ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   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:  lcvntr  34313  lcvnbtwn2  34314  lcvnbtwn3  34315