Theorem sh0le 28299

Description: The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
sh0le	|- ( A e. SH -> 0H C_ A )

Proof of Theorem sh0le

Step	Hyp	Ref	Expression
1		df-ch0 28110	. 2 |- 0H = { 0h }
2		sh0 28073	. . 3 |- ( A e. SH -> 0h e. A )
3	2	snssd 4340	. 2 |- ( A e. SH -> { 0h } C_ A )
4	1, 3	syl5eqss 3649	1 |- ( A e. SH -> 0H C_ A )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   {csn 4177   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

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-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:  ch0le  28300  shle0  28301  orthin  28305  ssjo  28306  shs0i  28308  span0  28401