Theorem chsspwh

Description: Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chsspwh	$$⊢{\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		chsssh	$$⊢{\mathbf{C}}_{\mathscr{H}}\subseteq {\mathbf{S}}_{\mathscr{H}}$$
2		shsspwh	$$⊢{\mathbf{S}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}$$
3	1 2	sstri	$$⊢{\mathbf{C}}_{\mathscr{H}}\subseteq 𝒫\mathscr{H}$$