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

Theorem elch0 22761
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 22760 . . 3  |-  0H  =  { 0h }
21eleq2i 2502 . 2  |-  ( A  e.  0H  <->  A  e.  { 0h } )
3 ax-hv0cl 22511 . . . 4  |-  0h  e.  ~H
43elexi 2967 . . 3  |-  0h  e.  _V
54elsnc2 3845 . 2  |-  ( A  e.  { 0h }  <->  A  =  0h )
62, 5bitri 242 1  |-  ( A  e.  0H  <->  A  =  0h )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   {csn 3816   ~Hchil 22427   0hc0v 22432   0Hc0h 22443
This theorem is referenced by:  ocin  22803  ocnel  22805  shuni  22807  choc0  22833  choc1  22834  omlsilem  22909  pjoc1i  22938  shne0i  22955  h1dn0  23059  spansnm0i  23157  nonbooli  23158  eleigvec  23465  cdjreui  23940  cdj3lem1  23942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-hv0cl 22511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-sn 3822  df-ch0 22760
  Copyright terms: Public domain W3C validator