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Theorem sh0le 21979
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 21792 . 2  |-  0H  =  { 0h }
2 sh0 21755 . . 3  |-  ( A  e.  SH  ->  0h  e.  A )
32snssd 3734 . 2  |-  ( A  e.  SH  ->  { 0h }  C_  A )
41, 3syl5eqss 3197 1  |-  ( A  e.  SH  ->  0H  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621    C_ wss 3127   {csn 3614   0hc0v 21464   SHcsh 21468   0Hc0h 21475
This theorem is referenced by:  ch0le  21980  shle0  21981  orthin  21985  ssjo  21986  shs0i  21988  span0  22081
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-hilex 21539
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-sh 21746  df-ch0 21792
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