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Theorem elch0 26599
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 26598 . . 3  |-  0H  =  { 0h }
21eleq2i 2482 . 2  |-  ( A  e.  0H  <->  A  e.  { 0h } )
3 ax-hv0cl 26347 . . . 4  |-  0h  e.  ~H
43elexi 3071 . . 3  |-  0h  e.  _V
54elsnc2 4005 . 2  |-  ( A  e.  { 0h }  <->  A  =  0h )
62, 5bitri 251 1  |-  ( A  e.  0H  <->  A  =  0h )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    = wceq 1407    e. wcel 1844   {csn 3974   ~Hchil 26263   0hc0v 26268   0Hc0h 26279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-hv0cl 26347
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-sn 3975  df-ch0 26598
This theorem is referenced by:  ocin  26641  ocnel  26643  shuni  26645  choc0  26671  choc1  26672  omlsilem  26747  pjoc1i  26776  shne0i  26793  h1dn0  26897  spansnm0i  26995  nonbooli  26996  eleigvec  27302  cdjreui  27777  cdj3lem1  27779
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