HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chle0 Unicode version

Theorem chle0 22022
Description: No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
Assertion
Ref Expression
chle0  |-  ( A  e.  CH  ->  ( A  C_  0H  <->  A  =  0H ) )

Proof of Theorem chle0
StepHypRef Expression
1 chsh 21804 . 2  |-  ( A  e.  CH  ->  A  e.  SH )
2 shle0 22021 . 2  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
31, 2syl 15 1  |-  ( A  e.  CH  ->  ( A  C_  0H  <->  A  =  0H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    C_ wss 3152   SHcsh 21508   CHcch 21509   0Hc0h 21515
This theorem is referenced by:  chle0i  22031  chssoc  22075  hatomistici  22942  atcvat4i  22977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-sh 21786  df-ch 21801  df-ch0 21832
  Copyright terms: Public domain W3C validator