Theorem shne0i 28307

Description: A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
shne0.1	|- A e. SH

Assertion

Ref	Expression
shne0i	|- ( A =/= 0H <-> E. x e. A x =/= 0h )

Distinct variable group:   x, A

Proof of Theorem shne0i

Step	Hyp	Ref	Expression
1		df-ne 2795	. 2 |- ( A =/= 0H <-> -. A = 0H )
2		df-rex 2918	. . 3 |- ( E. x e. A -. x e. 0H <-> E. x ( x e. A /\ -. x e. 0H ) )
3		nss 3663	. . 3 |- ( -. A C_ 0H <-> E. x ( x e. A /\ -. x e. 0H ) )
4		shne0.1	. . . . 5 |- A e. SH
5		shle0 28301	. . . . 5 |- ( A e. SH -> ( A C_ 0H <-> A = 0H ) )
6	4, 5	ax-mp 5	. . . 4 |- ( A C_ 0H <-> A = 0H )
7	6	notbii 310	. . 3 |- ( -. A C_ 0H <-> -. A = 0H )
8	2, 3, 7	3bitr2ri 289	. 2 |- ( -. A = 0H <-> E. x e. A -. x e. 0H )
9		elch0 28111	. . . 4 |- ( x e. 0H <-> x = 0h )
10	9	necon3bbii 2841	. . 3 |- ( -. x e. 0H <-> x =/= 0h )
11	10	rexbii 3041	. 2 |- ( E. x e. A -. x e. 0H <-> E. x e. A x =/= 0h )
12	1, 8, 11	3bitri 286	1 |- ( A =/= 0H <-> E. x e. A x =/= 0h )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   0hc0v 27781   SHcsh 27785   0Hc0h 27792

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-hilex 27856  ax-hv0cl 27860

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-sh 28064  df-ch0 28110

This theorem is referenced by:  chne0i  28312  shatomici  29217