Theorem chle0i 31354

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 31345	. 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ⊆ wss 3911   C_ℋ cch 30831  0_ℋc0h 30837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-hilex 30901

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fv 6507  df-ov 7372  df-sh 31109  df-ch 31123  df-ch0 31155

This theorem is referenced by:  chj00i  31389  chsup0  31450  spansnm0i  31552  largei  32169