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Theorem sh0le 28529
Description: The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
sh0le (𝐴S → 0𝐴)

Proof of Theorem sh0le
StepHypRef Expression
1 df-ch0 28340 . 2 0 = {0}
2 sh0 28303 . . 3 (𝐴S → 0𝐴)
32snssd 4448 . 2 (𝐴S → {0} ⊆ 𝐴)
41, 3syl5eqss 3755 1 (𝐴S → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2103  wss 3680  {csn 4285  0c0v 28011   S csh 28015  0c0h 28022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-hilex 28086
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-sh 28294  df-ch0 28340
This theorem is referenced by:  ch0le  28530  shle0  28531  orthin  28535  ssjo  28536  shs0i  28538  span0  28631
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