Theorem elch0 31587

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 31586	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2855	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31336	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3477	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4632	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 278	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {csn 4590   ℋchba 31252  0_ℎc0v 31257  0_ℋc0h 31268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-hv0cl 31336

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-sn 4591  df-ch0 31586

This theorem is referenced by:  ocin  31629  ocnel  31631  shuni  31633  choc0  31659  choc1  31660  omlsilem  31735  pjoc1i  31764  shne0i  31781  h1dn0  31885  spansnm0i  31983  nonbooli  31984  eleigvec  32290  cdjreui  32765  cdj3lem1  32767