Theorem elch0 31743

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 31742	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2854	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31492	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3475	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4629	. 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 2145  {csn 4587   ℋchba 31408  0_ℎc0v 31413  0_ℋc0h 31424

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 2147  ax-9 2155  ax-ext 2734  ax-hv0cl 31492

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sn 4588  df-ch0 31742

This theorem is used by:  ocin  31785  ocnel  31787  shuni  31789  choc0  31815  choc1  31816  omlsilem  31891  pjoc1i  31920  shne0i  31937  h1dn0  32041  spansnm0i  32139  nonbooli  32140  eleigvec  32446  cdjreui  32921  cdj3lem1  32923