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Theorem sh0le 21849
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 21662 . 2  |-  0H  =  { 0h }
2 sh0 21625 . . 3  |-  ( A  e.  SH  ->  0h  e.  A )
32snssd 3660 . 2  |-  ( A  e.  SH  ->  { 0h }  C_  A )
41, 3syl5eqss 3143 1  |-  ( A  e.  SH  ->  0H  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621    C_ wss 3078   {csn 3544   0hc0v 21334   SHcsh 21338   0Hc0h 21345
This theorem is referenced by:  ch0le  21850  shle0  21851  orthin  21855  ssjo  21856  shs0i  21858  span0  21951
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 1926  ax-ext 2234  ax-sep 4038  ax-hilex 21409
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 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-sh 21616  df-ch0 21662
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