Theorem elch0 31643

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 31642	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2858	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31392	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3480	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4636	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 278	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146  {csn 4594   ℋchba 31308  0_ℎc0v 31313  0_ℋc0h 31324

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-hv0cl 31392

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-sn 4595  df-ch0 31642

This theorem is used by:  ocin  31685  ocnel  31687  shuni  31689  choc0  31715  choc1  31716  omlsilem  31791  pjoc1i  31820  shne0i  31837  h1dn0  31941  spansnm0i  32039  nonbooli  32040  eleigvec  32346  cdjreui  32821  cdj3lem1  32823