Theorem sh0le

Description: The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	sh0le	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to 0_{\mathscr{H}}\subseteq A$$

Proof

Step	Hyp	Ref	Expression
1		df-ch0	$$⊢0_{\mathscr{H}}=\left\{0_{ℎ}\right\}$$
2		sh0	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to 0_{ℎ}\in A$$
3	2	snssd	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \left\{0_{ℎ}\right\}\subseteq A$$
4	1 3	eqsstrid	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to 0_{\mathscr{H}}\subseteq A$$