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Theorem sh0le 22011
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
StepHypRef Expression
1 df-ch0 21824 . 2  |-  0H  =  { 0h }
2 sh0 21787 . . 3  |-  ( A  e.  SH  ->  0h  e.  A )
32snssd 3761 . 2  |-  ( A  e.  SH  ->  { 0h }  C_  A )
41, 3syl5eqss 3223 1  |-  ( A  e.  SH  ->  0H  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1685    C_ wss 3153   {csn 3641   0hc0v 21496   SHcsh 21500   0Hc0h 21507
This theorem is referenced by:  ch0le  22012  shle0  22013  orthin  22017  ssjo  22018  shs0i  22020  span0  22113
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-hilex 21571
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-sh 21778  df-ch0 21824
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