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Theorem elch0 21833
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 21832 . . 3  |-  0H  =  { 0h }
21eleq2i 2347 . 2  |-  ( A  e.  0H  <->  A  e.  { 0h } )
3 ax-hv0cl 21583 . . . 4  |-  0h  e.  ~H
43elexi 2797 . . 3  |-  0h  e.  _V
54elsnc2 3669 . 2  |-  ( A  e.  { 0h }  <->  A  =  0h )
62, 5bitri 240 1  |-  ( A  e.  0H  <->  A  =  0h )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   {csn 3640   ~Hchil 21499   0hc0v 21504   0Hc0h 21515
This theorem is referenced by:  ocin  21875  ocnel  21877  shuni  21879  choc0  21905  choc1  21906  omlsilem  21981  pjoc1i  22010  shne0i  22027  h1dn0  22131  spansnm0i  22229  nonbooli  22230  eleigvec  22537  cdjreui  23012  cdj3lem1  23014
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-hv0cl 21583
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2790  df-sn 3646  df-ch0 21832
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