Theorem elch0 31862

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 31861	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2852	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31611	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3472	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4622	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 278	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {csn 4584   ℋchba 31527  0_ℎc0v 31532  0_ℋc0h 31543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-hv0cl 31611

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sn 4585  df-ch0 31861

This theorem is used by:  ocin  31904  ocnel  31906  shuni  31908  choc0  31934  choc1  31935  omlsilem  32010  pjoc1i  32039  shne0i  32056  h1dn0  32160  spansnm0i  32258  nonbooli  32259  eleigvec  32565  cdjreui  33040  cdj3lem1  33042