Theorem elch0 29517

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 29516	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2830	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 29266	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3441	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4597	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 274	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {csn 4558   ℋchba 29182  0_ℎc0v 29187  0_ℋc0h 29198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-hv0cl 29266

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-sn 4559  df-ch0 29516

This theorem is referenced by:  ocin  29559  ocnel  29561  shuni  29563  choc0  29589  choc1  29590  omlsilem  29665  pjoc1i  29694  shne0i  29711  h1dn0  29815  spansnm0i  29913  nonbooli  29914  eleigvec  30220  cdjreui  30695  cdj3lem1  30697