Theorem elch0 29612

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 29611	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2832	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 29361	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3450	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4606	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 274	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2110  {csn 4567   ℋchba 29277  0_ℎc0v 29282  0_ℋc0h 29293

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-hv0cl 29361

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-sn 4568  df-ch0 29611

This theorem is referenced by:  ocin  29654  ocnel  29656  shuni  29658  choc0  29684  choc1  29685  omlsilem  29760  pjoc1i  29789  shne0i  29806  h1dn0  29910  spansnm0i  30008  nonbooli  30009  eleigvec  30315  cdjreui  30790  cdj3lem1  30792