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Theorem elch0 24778
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 24777 . . 3  |-  0H  =  { 0h }
21eleq2i 2526 . 2  |-  ( A  e.  0H  <->  A  e.  { 0h } )
3 ax-hv0cl 24526 . . . 4  |-  0h  e.  ~H
43elexi 3064 . . 3  |-  0h  e.  _V
54elsnc2 3992 . 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 1370    e. wcel 1757   {csn 3961   ~Hchil 24442   0hc0v 24447   0Hc0h 24458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-hv0cl 24526
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-v 3056  df-sn 3962  df-ch0 24777
This theorem is referenced by:  ocin  24820  ocnel  24822  shuni  24824  choc0  24850  choc1  24851  omlsilem  24926  pjoc1i  24955  shne0i  24972  h1dn0  25076  spansnm0i  25174  nonbooli  25175  eleigvec  25482  cdjreui  25957  cdj3lem1  25959
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