Theorem elch0 31350

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 31349	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2832	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31099	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3455	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4604	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 276	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 207   = wceq 1547   ∈ wcel 2119  {csn 4562   ℋchba 31015  0_ℎc0v 31020  0_ℋc0h 31031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-hv0cl 31099

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-sn 4563  df-ch0 31349

This theorem is referenced by:  ocin  31392  ocnel  31394  shuni  31396  choc0  31422  choc1  31423  omlsilem  31498  pjoc1i  31527  shne0i  31544  h1dn0  31648  spansnm0i  31746  nonbooli  31747  eleigvec  32053  cdjreui  32528  cdj3lem1  32530