Theorem elch0 29031

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 29030	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2904	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 28780	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3513	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4604	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 277	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114  {csn 4567   ℋchba 28696  0_ℎc0v 28701  0_ℋc0h 28712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-hv0cl 28780

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sn 4568  df-ch0 29030

This theorem is referenced by:  ocin  29073  ocnel  29075  shuni  29077  choc0  29103  choc1  29104  omlsilem  29179  pjoc1i  29208  shne0i  29225  h1dn0  29329  spansnm0i  29427  nonbooli  29428  eleigvec  29734  cdjreui  30209  cdj3lem1  30211