Theorem sh0le 10997

Description: The zero subspace is the smallest subspace.

Assertion

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

Proof of Theorem sh0le

Step	Hyp	Ref	Expression
1		sh0 10717	. . 3 |- (A e. SH -> 0h e. A)
2	1	snssd 3130	. 2 |- (A e. SH -> {0h} C_ A)
3		df-ch0 10758	. 2 |- 0H = {0h}
4	2, 3	syl5ss 2661	1 |- (A e. SH -> 0H C_ A)

Syntax hints:   -> wi 3   e. wcel 1300   C_ wss 2593  {csn 3044  0hc0v 10423  SHcsh 10429  0Hc0h 10436

This theorem is referenced by:  ch0le 10998  shle0 10999  orthin 11003  shs0i 11005  span0 11098

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-sn 3049  df-sh 10709  df-ch0 10758

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