HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  sh0le Structured version   Unicode version

Theorem sh0le 26653
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 26466 . 2  |-  0H  =  { 0h }
2 sh0 26428 . . 3  |-  ( A  e.  SH  ->  0h  e.  A )
32snssd 4114 . 2  |-  ( A  e.  SH  ->  { 0h }  C_  A )
41, 3syl5eqss 3483 1  |-  ( A  e.  SH  ->  0H  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1840    C_ wss 3411   {csn 3969   0hc0v 26136   SHcsh 26140   0Hc0h 26147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-hilex 26211
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-xp 4946  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-sh 26419  df-ch0 26466
This theorem is referenced by:  ch0le  26654  shle0  26655  orthin  26659  ssjo  26660  shs0i  26662  span0  26755
  Copyright terms: Public domain W3C validator