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Theorem elch0 29015
 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 29014 . . 3 0 = {0}
21eleq2i 2903 . 2 (𝐴 ∈ 0𝐴 ∈ {0})
3 ax-hv0cl 28764 . . . 4 0 ∈ ℋ
43elexi 3490 . . 3 0 ∈ V
54elsn2 4577 . 2 (𝐴 ∈ {0} ↔ 𝐴 = 0)
62, 5bitri 278 1 (𝐴 ∈ 0𝐴 = 0)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2115  {csn 4540   ℋchba 28680  0ℎc0v 28685  0ℋc0h 28696 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793  ax-hv0cl 28764 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-sn 4541  df-ch0 29014 This theorem is referenced by:  ocin  29057  ocnel  29059  shuni  29061  choc0  29087  choc1  29088  omlsilem  29163  pjoc1i  29192  shne0i  29209  h1dn0  29313  spansnm0i  29411  nonbooli  29412  eleigvec  29718  cdjreui  30193  cdj3lem1  30195
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