Theorem elch0 31283

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 31282	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2831	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31032	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3501	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4670	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2106  {csn 4631   ℋchba 30948  0_ℎc0v 30953  0_ℋc0h 30964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-hv0cl 31032

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sn 4632  df-ch0 31282

This theorem is referenced by:  ocin  31325  ocnel  31327  shuni  31329  choc0  31355  choc1  31356  omlsilem  31431  pjoc1i  31460  shne0i  31477  h1dn0  31581  spansnm0i  31679  nonbooli  31680  eleigvec  31986  cdjreui  32461  cdj3lem1  32463