Theorem elch0 27495

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 27494	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2680	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 27244	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3186	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4158	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 263	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {csn 4125   ℋchil 27160  0_ℎc0v 27165  0_ℋc0h 27176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-hv0cl 27244

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sn 4126  df-ch0 27494

This theorem is referenced by:  ocin  27539  ocnel  27541  shuni  27543  choc0  27569  choc1  27570  omlsilem  27645  pjoc1i  27674  shne0i  27691  h1dn0  27795  spansnm0i  27893  nonbooli  27894  eleigvec  28200  cdjreui  28675  cdj3lem1  28677