Theorem elch0 31889

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 31888	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31638	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3476	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4626	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 278	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {csn 4588   ℋchba 31554  0_ℎc0v 31559  0_ℋc0h 31570

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-hv0cl 31638

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sn 4589  df-ch0 31888

This theorem is used by:  ocin  31931  ocnel  31933  shuni  31935  choc0  31961  choc1  31962  omlsilem  32037  pjoc1i  32066  shne0i  32083  h1dn0  32187  spansnm0i  32285  nonbooli  32286  eleigvec  32592  cdjreui  33067  cdj3lem1  33069