Theorem lssne0 18951

Description: A nonzero subspace has a nonzero vector. (shne0i 28307 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)

Hypotheses

Ref	Expression
lss0cl.z	|- .0. = ( 0g ` W )
lss0cl.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
lssne0	|- ( X e. S -> ( X =/= { .0. } <-> E. y e. X y =/= .0. ) )

Distinct variable groups:   y, X   y, .0.

Allowed substitution hints:   S( y)   W( y)

Proof of Theorem lssne0

Step	Hyp	Ref	Expression
1		lss0cl.s	. . . . 5 |- S = ( LSubSp ` W )
2	1	lssn0 18941	. . . 4 |- ( X e. S -> X =/= (/) )
3		eqsn 4361	. . . 4 |- ( X =/= (/) -> ( X = { .0. } <-> A. y e. X y = .0. ) )
4	2, 3	syl 17	. . 3 |- ( X e. S -> ( X = { .0. } <-> A. y e. X y = .0. ) )
5		nne 2798	. . . . 5 |- ( -. y =/= .0. <-> y = .0. )
6	5	ralbii 2980	. . . 4 |- ( A. y e. X -. y =/= .0. <-> A. y e. X y = .0. )
7		ralnex 2992	. . . 4 |- ( A. y e. X -. y =/= .0. <-> -. E. y e. X y =/= .0. )
8	6, 7	bitr3i 266	. . 3 |- ( A. y e. X y = .0. <-> -. E. y e. X y =/= .0. )
9	4, 8	syl6rbb 277	. 2 |- ( X e. S -> ( -. E. y e. X y =/= .0. <-> X = { .0. } ) )
10	9	necon1abid 2832	1 |- ( X e. S -> ( X =/= { .0. } <-> E. y e. X y =/= .0. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   {csn 4177   ` cfv 5888   0gc0g 16100   LSubSpclss 18932

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

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-ov 6653  df-lss 18933

This theorem is referenced by:  lsmsat  34295  lssatomic  34298  dochsatshpb  36741  hgmapvvlem3  37217