Theorem chle0i 31476

Description: No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ch0le.1	⊢ 𝐴 ∈ Cℋ

Assertion

Ref	Expression
chle0i	⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)

Proof of Theorem chle0i

Step	Hyp	Ref	Expression
1		ch0le.1	. 2 ⊢ 𝐴 ∈ Cℋ
2		chle0 31467	. 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113   ⊆ wss 3899   C_ℋ cch 30953  0_ℋc0h 30959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-hilex 31023

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fv 6498  df-ov 7359  df-sh 31231  df-ch 31245  df-ch0 31277

This theorem is referenced by:  chj00i  31511  chsup0  31572  spansnm0i  31674  largei  32291