HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  elch0 Structured version   Unicode version

Theorem elch0 25848
Description: Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
Assertion
Ref Expression
elch0  |-  ( A  e.  0H  <->  A  =  0h )

Proof of Theorem elch0
StepHypRef Expression
1 df-ch0 25847 . . 3  |-  0H  =  { 0h }
21eleq2i 2545 . 2  |-  ( A  e.  0H  <->  A  e.  { 0h } )
3 ax-hv0cl 25596 . . . 4  |-  0h  e.  ~H
43elexi 3123 . . 3  |-  0h  e.  _V
54elsnc2 4058 . 2  |-  ( A  e.  { 0h }  <->  A  =  0h )
62, 5bitri 249 1  |-  ( A  e.  0H  <->  A  =  0h )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   {csn 4027   ~Hchil 25512   0hc0v 25517   0Hc0h 25528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-hv0cl 25596
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sn 4028  df-ch0 25847
This theorem is referenced by:  ocin  25890  ocnel  25892  shuni  25894  choc0  25920  choc1  25921  omlsilem  25996  pjoc1i  26025  shne0i  26042  h1dn0  26146  spansnm0i  26244  nonbooli  26245  eleigvec  26552  cdjreui  27027  cdj3lem1  27029
  Copyright terms: Public domain W3C validator