Theorem ch0le

Description: The zero subspace is the smallest member of CH . (Contributed by NM, 14-Aug-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	ch0le	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to 0_{\mathscr{H}}\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		chsh	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to A\in {\mathbf{S}}_{\mathscr{H}}$$
2		sh0le	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to 0_{\mathscr{H}}\subseteq A$$
3	1 2	syl	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to 0_{\mathscr{H}}\subseteq A$$