Theorem chle0i 31655

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 31646	. 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)

Syntax hints:   ↔ wb 208   = wceq 1560   ∈ wcel 2142   ⊆ wss 3904   C_ℋ cch 31132  0_ℋc0h 31138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-hilex 31202

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fv 6529  df-ov 7399  df-sh 31410  df-ch 31424  df-ch0 31456

This theorem is referenced by:  chj00i  31690  chsup0  31751  spansnm0i  31853  largei  32470