Theorem chle0i

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

	Ref	Expression
Hypothesis	ch0le.1	⊢ 𝐴 ∈ Cℋ
Assertion	chle0i	⊢ ( 𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ )

Proof

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