Theorem chle0i 32066

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 32057	. 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2146   ⊆ wss 3899   C_ℋ cch 31543  0_ℋc0h 31549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-hilex 31613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fv 6554  df-ov 7430  df-sh 31821  df-ch 31835  df-ch0 31867

This theorem is used by:  chj00i  32101  chsup0  32162  spansnm0i  32264  largei  32881