Theorem elch0 31413

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 31412	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2853	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31162	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3475	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4621	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 277	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 208   = wceq 1559   ∈ wcel 2141  {csn 4579   ℋchba 31078  0_ℎc0v 31083  0_ℋc0h 31094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-hv0cl 31162

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-sn 4580  df-ch0 31412

This theorem is referenced by:  ocin  31455  ocnel  31457  shuni  31459  choc0  31485  choc1  31486  omlsilem  31561  pjoc1i  31590  shne0i  31607  h1dn0  31711  spansnm0i  31809  nonbooli  31810  eleigvec  32116  cdjreui  32591  cdj3lem1  32593