Theorem elch0 29037

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 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Proof of Theorem elch0

Step	Hyp	Ref	Expression
1		df-ch0 29036	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2881	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 28786	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3460	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4564	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 278	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {csn 4525   ℋchba 28702  0_ℎc0v 28707  0_ℋc0h 28718

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-hv0cl 28786

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sn 4526  df-ch0 29036

This theorem is referenced by:  ocin  29079  ocnel  29081  shuni  29083  choc0  29109  choc1  29110  omlsilem  29185  pjoc1i  29214  shne0i  29231  h1dn0  29335  spansnm0i  29433  nonbooli  29434  eleigvec  29740  cdjreui  30215  cdj3lem1  30217