Theorem elch0 29616

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 29615	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2830	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 29365	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3451	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4600	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 274	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {csn 4561   ℋchba 29281  0_ℎc0v 29286  0_ℋc0h 29297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-hv0cl 29365

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sn 4562  df-ch0 29615

This theorem is referenced by:  ocin  29658  ocnel  29660  shuni  29662  choc0  29688  choc1  29689  omlsilem  29764  pjoc1i  29793  shne0i  29810  h1dn0  29914  spansnm0i  30012  nonbooli  30013  eleigvec  30319  cdjreui  30794  cdj3lem1  30796