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Theorem elch0 28341
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 28340 . . 3 0 = {0}
21eleq2i 2795 . 2 (𝐴 ∈ 0𝐴 ∈ {0})
3 ax-hv0cl 28090 . . . 4 0 ∈ ℋ
43elexi 3317 . . 3 0 ∈ V
54elsn2 4319 . 2 (𝐴 ∈ {0} ↔ 𝐴 = 0)
62, 5bitri 264 1 (𝐴 ∈ 0𝐴 = 0)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1596  wcel 2103  {csn 4285  chil 28006  0c0v 28011  0c0h 28022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-hv0cl 28090
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-sn 4286  df-ch0 28340
This theorem is referenced by:  ocin  28385  ocnel  28387  shuni  28389  choc0  28415  choc1  28416  omlsilem  28491  pjoc1i  28520  shne0i  28537  h1dn0  28641  spansnm0i  28739  nonbooli  28740  eleigvec  29046  cdjreui  29521  cdj3lem1  29523
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