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Theorem shle0 26022
Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shle0  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )

Proof of Theorem shle0
StepHypRef Expression
1 sh0le 26020 . . 3  |-  ( A  e.  SH  ->  0H  C_  A )
21biantrud 507 . 2  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  ( A  C_  0H  /\  0H  C_  A ) ) )
3 eqss 3512 . 2  |-  ( A  =  0H  <->  ( A  C_  0H  /\  0H  C_  A ) )
42, 3syl6bbr 263 1  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3469   SHcsh 25507   0Hc0h 25514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-hilex 25578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-sh 25786  df-ch0 25833
This theorem is referenced by:  chle0  26023  shne0i  26028  shs00i  26030  cdj3lem1  27015
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