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Theorem sh0le 22074
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 21887 . 2  |-  0H  =  { 0h }
2 sh0 21850 . . 3  |-  ( A  e.  SH  ->  0h  e.  A )
32snssd 3797 . 2  |-  ( A  e.  SH  ->  { 0h }  C_  A )
41, 3syl5eqss 3256 1  |-  ( A  e.  SH  ->  0H  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1701    C_ wss 3186   {csn 3674   0hc0v 21559   SHcsh 21563   0Hc0h 21570
This theorem is referenced by:  ch0le  22075  shle0  22076  orthin  22080  ssjo  22081  shs0i  22083  span0  22176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-hilex 21634
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-cnv 4734  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-sh 21841  df-ch0 21887
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