Theorem elch0 31236

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 31235	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2825	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 30985	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3460	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4617	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {csn 4575   ℋchba 30901  0_ℎc0v 30906  0_ℋc0h 30917

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-hv0cl 30985

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-sn 4576  df-ch0 31235

This theorem is referenced by:  ocin  31278  ocnel  31280  shuni  31282  choc0  31308  choc1  31309  omlsilem  31384  pjoc1i  31413  shne0i  31430  h1dn0  31534  spansnm0i  31632  nonbooli  31633  eleigvec  31939  cdjreui  32414  cdj3lem1  32416