Theorem shle0

Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shle0	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \left(A\subseteq 0_{\mathscr{H}}\leftrightarrow A=0_{\mathscr{H}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		sh0le	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to 0_{\mathscr{H}}\subseteq A$$
2	1	biantrud	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \left(A\subseteq 0_{\mathscr{H}}\leftrightarrow \left(A\subseteq 0_{\mathscr{H}}\wedge 0_{\mathscr{H}}\subseteq A\right)\right)$$
3		eqss	$$⊢A=0_{\mathscr{H}}\leftrightarrow \left(A\subseteq 0_{\mathscr{H}}\wedge 0_{\mathscr{H}}\subseteq A\right)$$
4	2 3	bitr4di	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \left(A\subseteq 0_{\mathscr{H}}\leftrightarrow A=0_{\mathscr{H}}\right)$$