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Theorem elch0 28958
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 ∈ 0𝐴 = 0)

Proof of Theorem elch0
StepHypRef Expression
1 df-ch0 28957 . . 3 0 = {0}
21eleq2i 2901 . 2 (𝐴 ∈ 0𝐴 ∈ {0})
3 ax-hv0cl 28707 . . . 4 0 ∈ ℋ
43elexi 3511 . . 3 0 ∈ V
54elsn2 4594 . 2 (𝐴 ∈ {0} ↔ 𝐴 = 0)
62, 5bitri 276 1 (𝐴 ∈ 0𝐴 = 0)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  {csn 4557  chba 28623  0c0v 28628  0c0h 28639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-hv0cl 28707
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-sn 4558  df-ch0 28957
This theorem is referenced by:  ocin  29000  ocnel  29002  shuni  29004  choc0  29030  choc1  29031  omlsilem  29106  pjoc1i  29135  shne0i  29152  h1dn0  29256  spansnm0i  29354  nonbooli  29355  eleigvec  29661  cdjreui  30136  cdj3lem1  30138
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