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Theorem sh0le 25015
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 24828 . 2  |-  0H  =  { 0h }
2 sh0 24790 . . 3  |-  ( A  e.  SH  ->  0h  e.  A )
32snssd 4129 . 2  |-  ( A  e.  SH  ->  { 0h }  C_  A )
41, 3syl5eqss 3511 1  |-  ( A  e.  SH  ->  0H  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    C_ wss 3439   {csn 3988   0hc0v 24498   SHcsh 24502   0Hc0h 24509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-hilex 24573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-cnv 4959  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-sh 24781  df-ch0 24828
This theorem is referenced by:  ch0le  25016  shle0  25017  orthin  25021  ssjo  25022  shs0i  25024  span0  25117
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